eess.SYFeb 25, 2026

The Port-Hamiltonian Structure of Vehicle Manipulator Systems

Authors: Ramy Rashad

Organizations: Control and Instrumentation Engineering department and Interdisciplinary Research Center for Smart Mobility and Logistics, King Fahd University of Petroleum and Minerals, Dhahran, 34464, Saudi Arabia

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Abstract

This paper presents a port-Hamiltonian formulation of vehicle-manipulator systems (VMS), a broad class of robotic systems including aerial manipulators, underwater manipulators, space robots, and omnidirectional mobile manipulators. Unlike existing Lagrangian formulations that obscure the underlying energetic structure, the proposed port-Hamiltonian formulation explicitly reveals the energy flow and conservation properties of these complex mechanical systems. We derive the port-Hamiltonian dynamics from first principles using Hamiltonian reduction theory. Two complementary formulations are presented: a standard form that directly exposes the energy structure, and an inertially-decoupled form that leverages the principal bundle structure of the VMS configuration space and is particularly suitable for control design and numerical simulation. The coordinate-free geometric approach we follow avoids singularities associated with local parameterizations of the base orientation. We rigorously establish the mathematical equivalence between our port-Hamiltonian formulations and existing reduced Euler-Lagrange and Boltzmann-Hamel equations found in the robotics and geometric mechanics literature.

Figures & tables

Figure 1: Illustration of vehicle-manipulator systems. Left figure shows a floating-base manipulator h∈Gb=SE(3)h\in G_{b}=SE(3) while the right figure shows a ground mobile manipulator h∈Gb=S1×R2h\in G_{b}=\mathbb{S}^{1}\times\mathbb{R}^{2} .
Map/VariableCoordinate-freeCoordinate-based
(v,q˙)(\text{v},\dot{q})gb×TqQm\mathfrak{g}_{b}\times T_{q}Q_{m}Rb×Rn\mathbb{R}^{b}\times\mathbb{R}^{n}
(w,τ)(\text{w},\tau)gb∗×Tq∗Qm\mathfrak{g}_{b}^{*}\times T_{q}^{*}Q_{m}Rb×Rn\mathbb{R}^{b}\times\mathbb{R}^{n}
Jii,s(q)J_{i}^{i,s}(q)gb×TqQm→gis\mathfrak{g}_{b}\times T_{q}Q_{m}\rightarrow\mathfrak{g}_{i}^{s}R6×(b+n)\mathbb{R}^{6\times(b+n)}
Jii,b(q)J_{i}^{i,b}(q)TqQm→gibT_{q}Q_{m}\rightarrow\mathfrak{g}_{i}^{b}R6×n\mathbb{R}^{6\times n}
Mb(q)M_{b}(q)gb→gb∗\mathfrak{g}_{b}\rightarrow{\mathfrak{g}_{b}^{*}}Rb×b\mathbb{R}^{b\times b}
Mbm(q)M_{bm}(q)TqQm→gb∗T_{q}Q_{m}\rightarrow{\mathfrak{g}_{b}^{*}}Rb×n\mathbb{R}^{b\times n}
Table 1: Coordinate-free and coordinate-based representations of maps and variables in ( 21 )
Figure 2: Port-Hamiltonian formulation of fixed-base manipulator dynamics.
Figure 3: Port-Hamiltonian formulation of moving-base dynamics.
Map/VariableCoordinate-freeCoordinate-based
(p˙,∂Hkin∂p)(\dot{\text{p}},\frac{\partial\mathcal{H}_{\text{kin}}}{\partial\text{p}})gb∗×gb\mathfrak{g}_{b}^{*}\times\mathfrak{g}_{b}Rb×Rb\mathbb{R}^{b}\times\mathbb{R}^{b}
(q˙,∂Hkin∂q)(\dot{q},\frac{\partial\mathcal{H}_{\text{kin}}}{\partial q})TqQm×Tq∗QmT_{q}Q_{m}\times T_{q}^{*}Q_{m}Rn×Rn\mathbb{R}^{n}\times\mathbb{R}^{n}
(π˙,∂Hkin∂π)(\dot{\pi},\frac{\partial\mathcal{H}_{\text{kin}}}{\partial\pi})Tq∗Qm×TqQmT_{q}^{*}Q_{m}\times T_{q}Q_{m}Rn×Rn\mathbb{R}^{n}\times\mathbb{R}^{n}
adp∼ad_{\text{p}}^{\sim}gb→gb∗\mathfrak{g}_{b}\rightarrow\mathfrak{g}_{b}^{*}Rb×b\mathbb{R}^{b\times b}
J(p)\mathcal{J}(\text{p})Tx∗X→TxXT_{x}^{*}\mathcal{X}\rightarrow T_{x}\mathcal{X}R(b+2n)×(b+2n)\mathbb{R}^{(b+2n)\times(b+2n)}
G\mathcal{G}gb∗×Tq∗Qm→TxX\mathfrak{g}_{b}^{*}\times T_{q}^{*}Q_{m}\rightarrow T_{x}\mathcal{X}R(b+2n)×(b+n)\mathbb{R}^{(b+2n)\times(b+n)}
Table 2: Coordinate-free and coordinate-based representations of maps and variables in ( 47 )
Figure 4: Principal bundle structure of the VMS configuration space Q=Gb×QmQ=G_{b}\times Q_{m} .
MapCoordinate-freeCoordinate-based
A(q)A(q)TqQm→gb{T_{q}Q_{m}}\rightarrow{\mathfrak{g}_{b}}Rb×n\mathbb{R}^{b\times n}
Φ(p,q)\Phi(\text{p},q)TxX→Tx^XT_{x}\mathcal{X}\rightarrow T_{\hat{x}}\mathcal{X}R(b+2n)×(b+2n)\mathbb{R}^{(b+2n)\times(b+2n)}
L(p,q)L(\text{p},q)TqQm→Tq∗QmT_{q}Q_{m}\rightarrow T_{q}^{*}Q_{m}Rn×n\mathbb{R}^{n\times n}
Nb(p^,q^)N_{b}(\hat{\text{p}},\hat{q})Tq^Qm→gbT_{\hat{q}}Q_{m}\rightarrow\mathfrak{g}_{b}Rb×n\mathbb{R}^{b\times n}
N^m(q^,π^)\hat{N}_{m}(\hat{q},\hat{\pi})Tq^Qm→Tq^∗QmT_{\hat{q}}Q_{m}\rightarrow T_{\hat{q}}^{*}Q_{m}Rn×n\mathbb{R}^{n\times n}
B(p^,q^)B(\hat{\text{p}},\hat{q})Tq^Qm→Tq^∗Qm{T_{\hat{q}}Q_{m}}\rightarrow{T_{\hat{q}}^{*}Q_{m}}Rn×n\mathbb{R}^{n\times n}
Table 3: Coordinate-free and coordinate-based representations of maps in ( 59 )
Figure 5: The proposed port-Hamiltonian formulations of VMS dynamics and their corresponding Dirac structures. Left figure shows the port-Hamiltonian formulation ( 47 ) while the right figure shows the inertially-decoupled port-Hamiltonian formulation ( 59 ). The intermediate figure illustrates the change of coordinates ϕ(q)\phi(q) .
MapCoordinate-freeCoordinate-based
Pb(q,q˙)P_{b}(q,\dot{q})gb→gb{\mathfrak{g}_{b}}\rightarrow{\mathfrak{g}_{b}}Rb×b\mathbb{R}^{b\times b}
Eb(v^,q)E_{b}(\hat{\text{v}},q)TqQm→gb{T_{q}Q_{m}}\rightarrow{\mathfrak{g}_{b}}Rb×n\mathbb{R}^{b\times n}
C^m(q,q˙)\hat{C}_{m}(q,\dot{q})TqQm→Tq∗Qm{T_{q}Q_{m}}\rightarrow{T_{q}^{*}Q_{m}}Rn×n\mathbb{R}^{n\times n}
B^(v^,q^)\hat{B}(\hat{\text{v}},\hat{q})Tq^Qm→Tq^∗Qm{T_{\hat{q}}Q_{m}}\rightarrow{T_{\hat{q}}^{*}Q_{m}}Rn×n\mathbb{R}^{n\times n}
M^,C^1,C^2\hat{\mathcal{M}},\hat{\mathcal{C}}_{1},\hat{\mathcal{C}}_{2}gb×TqQm→gb∗×Tq∗Qm{\mathfrak{g}_{b}\times T_{q}Q_{m}}\rightarrow{\mathfrak{g}_{b}^{*}\times T_{q}^{*}Q_{m}}R(b+n)×(b+n)\mathbb{R}^{(b+n)\times(b+n)}
Table 4: Coordinate-free and coordinate-based representations of maps in ( 85 )
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