Passivity-preserving splitting methods for rigid body systems

Abstract

A rigid body model for the dynamics of a marine vessel, used in simulations of offshore pipe-lay operations, gives rise to a set of ordinary differential equations with controls. The system is input–output passive. We propose passivity-preserving splitting methods for the numerical solution of a class of problems which includes this system as a special case. We prove the passivity-preservation property for the splitting methods, and we investigate stability and energy behaviour in numerical experiments. Implementation is discussed in detail for a special case where the splitting gives rise to the subsequent integration of two completely integrable flows. The equations for the attitude are reformulated on $\mathit{SO}(3)$ using rotation matrices rather than local parameterisations with Euler angles.

Appendices

Appendix A: Euler parameters

We here review briefly the main properties of quaternions and Euler parameters. More information on this subject can be found, e.g. in [20]. The set of quaternions,

$$ \mathbb{H}:=\bigl\{ q=(q_{0},\mathbf{q})\in\mathbb{R}\times \mathbb{R}^{3}, \, \mathbf{q}=[q_{1},q_{2},q_{3}]^{T} \bigr\} \cong\mathbb{R}^{4}, $$

is a strictly skew field [1]. Addition and multiplication of two quaternions, $p=(p_{0},\mathbf{p})$, $q=(q_{0},\mathbf{q}) \in\mathbb{H}$, are defined by

$$ p+q:=(p_{0}+q_{0},\mathbf{p}+\mathbf{q}), $$

and

$$ pq:=\bigl(p_{0}q_{0}-\mathbf{p}^{T} \mathbf{q},p_{0}\mathbf{q}+q_{0} \mathbf{p}+\mathbf{p} \times\mathbf{q}\bigr). $$

(A.1)

For $q\neq(0,\mathbf{0})$ there exists an inverse

$$ q^{-1}:=q^{c}/\| q\|^{2},\quad\|q\| := \sqrt {q_{0}^{2} + \| \mathbf{q}\|_{2}^{2}}, $$

where $q^{c}:=(q_{0},-\mathbf{q})$ is the conjugate of $q$, such that $qq^{-1}=q^{-1}q= e=(1,\mathbf{0})$. In the sequel we will consider $q\in\mathbb{H}$ as a vector $q=[q_{0},q_{1},q_{2},q_{3}]^{T}\in \mathbb{R}^{4}$. The multiplication rule (A.1) can then be expressed by means of a matrix–vector product in $\mathbb{R}^{4}$. Namely, $pq=L(p)q=R(q)p$, where

$$ L(p):=\left [ \textstyle\begin{array}{c@{\quad}c} p_{0}&-\mathbf{p}^{T} \\ \mathbf{p} & p_{0}I+\widehat{\mathbf{p}} \end{array}\displaystyle \right ] , \qquad R(q):=\left [ \textstyle\begin{array}{c@{\quad}c} q_{0}&-\mathbf{q}^{T} \\ \mathbf{q} & q_{0}I-\widehat{\mathbf{q}} \end{array}\displaystyle \right ] , $$

and $I$ is the $3\times3$ identity matrix. Note that $R(q)$ and $L(p)$ commute, i.e. $R(q)L(p)=L(p)R(q)$.

Three-dimensional rotations in space can be represented by Euler parameters, i.e. unit quaternions

$$ \mathbb{S}^{3}:=\{q \in\mathbb{H}\, | \, \|q\|=1 \}. $$

Equipped with the quaternion product, $\mathbb{S}^{3}$ is a Lie group, with $q^{-1}=q^{c}$ and $e = (1,\mathbf{0})$ as the identity element. There exists a (surjective $2:1$) group homomorphism (the Euler–Rodriguez map) $\mathcal{E}:\mathbb{S}^{3} \to\mathit{SO}(3)$, defined by

$$ \mathcal{E}(q) := I +2q_{0}\widehat{\mathbf{q}}+2\widehat{\mathbf{q}} ^{2}. $$

The Euler–Rodriguez map can be explicitly written as

$$ \mathcal{E}(q) = \left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c} 1-2(q_{2}^{2}+q_{3}^{2})& 2(q_{1}q_{2}-q_{0}q_{3})& 2(q_{0}q_{2}+q _{1}q_{3}) \\ 2(q_{0}q_{3}+q_{1}q_{2})&1-2(q_{1}^{2}+q_{3}^{2})&2(q_{2}q_{3}-q_{0}q _{1}) \\ 2(q_{1}q_{3}-q_{0}q_{2})&2(q_{0}q_{1}+q_{2}q_{3})&1-2(q_{1}^{2}+q_{2} ^{2}) \end{array}\displaystyle \right ] . $$

A rotation in $\mathbb{R}^{3}$,

$$ \mathbf{w}=Q\mathbf{u},\quad Q\in\mathit{SO}(3), \quad\mathbf{u},\mathbf{w} \in \mathbb{R}^{3}, $$

can, for some $q\in\mathbb{S}^{3}$, be expressed in quaternionic form as

$$ w=L(q)R\bigl(q^{c}\bigr) u= R\bigl(q^{c} \bigr)L(q) u, \quad u=(0,\mathbf{u}),\ w=(0, \mathbf{w})\in\mathbb{H}_{\mathcal{P}}, $$

where $\mathbb{H}_{\mathcal{P}}:=\{q\in\mathbb{H}\, | \, q_{0}=0 \} \cong\mathbb{R}^{3}$ is the set of so called pure quaternions.

1.1 A.1 The Lie algebra $\mathfrak{s}^{3}$

If $q\in\mathbb{S}^{3}$, it follows from $qq^{c}=e$ that

$$ \mathfrak{s}^{3}:=T_{e}\mathbb{S}^{3}= \mathbb{H}_{\mathcal{P}}. $$

The Lie algebra $\mathfrak{s}^{3}$, associated with $\mathbb{S}^{3}$, is equipped with a Lie bracket $[\, \cdot\, ,\cdot\,]_{\mathfrak{s}}: \mathfrak{s}^{3}\times\mathfrak{s}^{3} \to\mathfrak{s}^{3}$,

$$ [\,u\, , w\,]_{\mathfrak{s}}:= L(u)w-L(w)u=(0,2 \mathbf{u}\times \mathbf{w}), $$

where $u=(0,\mathbf{u})$, $w=(0,\mathbf{w})$.

The derivative map of ℰ is $\mathcal{E}_{*} =T_{e} \mathcal{E}:\mathfrak{s}^{3}\to\mathfrak{so}(3)$. This map, given by

$$ \mathcal{E}_{*}(u)=2\widehat{\mathbf{u}},\quad u=(0, \mathbf{u}), $$

is a Lie algebra isomorphism. Assume now that $q\in\mathbb{S}^{3}$ is such that $\mathcal{E}(q(t))=Q(t)$, then $L(q^{c})\dot{q}\in \mathfrak{s}^{3}$, $Q^{T}\dot{Q}\in\mathfrak{so}(3)$ and

$$ \mathcal{E}_{*}\bigl(L\bigl(q^{c}\bigr) \dot{q}\bigr) = Q^{T}\dot{Q}. $$

(A.2)

Furthermore, it can be shown that

$$ \mathcal{E}_{*}\bigl(L(q)R\bigl(q^{c}\bigr) u \bigr) = 2 \widehat{\mathcal{E}(q)\mathbf{u}} \quad \forall q\in \mathbb{S}^{3} ,\, u=(0,\mathbf{u}) \in\mathfrak{s} ^{3}. $$

(A.3)

As a consequence of (A.2) and (A.3), the kinematics of the attitude of the vessel (23) can be expressed in Euler parameters in $\mathbb{S}^{3}$ as

$$ \dot{q} = \frac{1}{2}q\,\omega, \quad \omega=(0, \boldsymbol{\omega}). $$

(A.4)

Appendix B: The equations using Euler parameters

We rewrite the equations (31), (32), (33), (34), using (A.4) to represent the attitude with Euler parameters. Note that if $q \in \mathbb{S}^{3}$ is known, we also know the Euler angles ${\boldsymbol{\theta}}$ from a transformation between the two representations:

$$\begin{aligned} \dot{\mathbf{p}} &= \mathbf{p}\times\boldsymbol{\omega} - \bigl( D_{t}\mathbf{v}+\mathcal{E}(q)^{T}\mathbf{g}_{t}^{s}- \boldsymbol{\tau}_{t} \bigr) , \\ \dot{\mathbf{m}} &= \mathbf{m} \times\boldsymbol{\omega} + \mathbf{p}\times \mathbf{v}- \bigl( D_{r}\boldsymbol{\omega}+ \mathcal{E}(q)^{T} \mathbf{g}_{r}^{s}-{\boldsymbol{\tau}}_{r} \bigr) , \\ \dot{q} &= \frac{1}{2}q\,\omega, \quad \omega=(0,\boldsymbol{\omega}), \\ \dot{\mathbf{x}} &= \mathcal{E}(q)\,\mathbf{v}, \\ \dot{\boldsymbol{\varphi}}_{\boldsymbol{\theta}} &= \tilde{{\boldsymbol{\theta}}}, \\ \dot{\boldsymbol{\varphi}}_{\mathbf{x}} &=\tilde{\mathbf{x}}, \end{aligned}$$

with

$$\begin{aligned} {\boldsymbol{\tau}}_{r} &= - \bigl( \varPi_{e}^{-1} \mathcal{E}(q) \bigr) ^{T}\bigl(K_{p}^{r} \tilde{{ \boldsymbol{\theta}}} + K_{d}^{r} \dot{\tilde{{\boldsymbol{ \theta}}}} + K_{i}^{r} \boldsymbol{\varphi}_{\boldsymbol{\theta}} \bigr), \\ \boldsymbol{\tau}_{t} &= -\mathcal{E}(q)^{T} \bigl( K_{p}^{t} \tilde{\mathbf{x}} + K_{d}^{t} \dot{\tilde{\mathbf{x}}} + K_{i}^{t} \boldsymbol{ \varphi}_{\mathbf{x}} \bigr) , \\ \tilde{{\boldsymbol{\theta}}} &= {\boldsymbol{\theta}}- {\boldsymbol{ \theta}}_{\mathrm{ref}}, \hspace{6mm} \dot{\tilde{{\boldsymbol{\theta}}}} = \dot{{ \boldsymbol{\theta}}} = \varPi_{e}^{-1}\mathcal{E}(q)\, \boldsymbol{\omega} , \hspace{6mm} \tilde{\mathbf{x}} = \mathbf{x}- \mathbf{x}_{\mathrm{ref}}. \end{aligned}$$

Appendix C: Splitting coefficients

The coefficients of a 4th order splitting scheme in the format (10) are

$$\begin{aligned} a_{1} =& 0.0792036964311956500000000000000000000000, \\ a_{2} =& 0.353172906049773728818833445330, \\ a_{3} =& -0.042065080357719520000000000000000000000, \\ a_{4} =& 1-2(a_{1}+a_{2}+a_{3}), \\ b_{1} =& 0.209515106613361881525060713987, \\ b_{2} =& -0.14385177317981800000000000000000000, \\ b_{3} =& \frac{1}{2}-(b_{1}+b_{2}). \end{aligned}$$

The coefficients of a 6th order splitting scheme in the format (10) are

$$\begin{aligned} a_{1} =& 0.0502627644003923808654389538920, \\ a_{2} =& 0.413514300428346618921141630839, \\ a_{3} =& 0.045079889794397660000000000000000000, \\ a_{4} =& -0.188054853819571375656897886496, \\ a_{5} =& 0.541960678450781151905056284542, \\ a_{6} =& 1-2(a_{1}+a_{2}+a_{3}+a_{4}+a_{5}), \\ b_{1} =& 0.148816447901042828823498193483, \\ b_{2} =& -0.132385865767782744686048193902, \\ b_{3} =& 0.0673076046921849473963237618218, \\ b_{4} =& 0.432666402578172649872653897748, \\ b_{5} =& \frac {1}{2}-(b_{1}+b_{2}+b_{3}+b_{4}). \end{aligned}$$

We refer to [2] for an overview on splitting schemes.

Appendix D: Parameter values

The values of the parameters we use in the experiments are in SI units

$$\begin{aligned} \textstyle\begin{array}{rl@{\qquad}rl} D_{r1} & = 9.329153987 \times10^{2}, & D_{t1} & = 3.53933789 \times 10^{1}, \\ D_{r2} & = 6.514979127508227 \times10^{8}, & D_{t2} & = 1.1781388 \times10^{2}, \\ D_{r3} & = 3.15094664584 \times10^{4}, & D_{t3} & = 1.4566249 \times 10^{6}, \\ D_{r} & = \mathrm{diag}(D_{r1},D_{r2},D_{r3}), & D_{t} & = \mathrm {diag}(D_{t1},D_{t2},D_{t3}), \\ K_{p}^{r} & = \mathrm{diag}\bigl(0,0,1\cdot10^{8}\bigr), & K_{p}^{t} & = \mathrm {diag}\bigl(4\cdot10^{5},4\cdot10^{5},0\bigr), \\ K_{d}^{r} & = \mathrm{diag}\bigl(0,0,1\cdot10^{9}\bigr), & K_{d}^{t} & = \mathrm {diag}\bigl(4\cdot10^{6},4\cdot10^{6},0\bigr), \\ K_{i}^{r} & = \mathrm{diag}\bigl(0,0,2\cdot10^{5}\bigr), & K_{i}^{t} & = \mathrm {diag}\bigl(1\cdot10^{3},1\cdot10^{3},0\bigr), \\ T_{1} & = 2.873071 \times10^{8} , & {\boldsymbol{\theta}}_{0} & = [0.05,-0.02,0.10]^{T}, \\ T_{2} & = 2.726143\times10^{9} , & {\boldsymbol{\theta}}_{\mathrm{ref}} & = [ 0,0,0.54 ] , \\ T_{3} & = 2.90000 \times10^{9} , & \mathbf{x}_{0} & = [723,0,0]^{T}, \\ T & =\mathrm{diag}(T_{1},T_{2},T_{3}), & \mathbf{x}_{\mathrm{ref}} & = [ 780,20,0 ] , \\ \overline{GM}_{T} & = 2.1440 , & m_{v} & = 6.3622085\times10^{6}, \\ \overline{GM}_{L} & = 103.628 , & A_{\mathrm{wp}} & = 1.3834\times10^{3} , \\ g & = 9.81 , & \\ \rho_{w} & = 1.025 \times10^{3} , & \\ z_{\mathrm{eq}} & = 0. & \\ \end{array}\displaystyle \end{aligned}$$

Many of the values are taken from data for a supply vessel from [23].