Controlled Lagrangians and stabilization of Euler–Poincaré mechanical systems with broken symmetry II: potential shaping

Abstract

We apply the method of controlled Lagrangians by potential shaping to Euler–Poincaré mechanical systems with broken symmetry. We assume that the configuration space is a general semidirect product Lie group ${\mathsf {G}} \ltimes V$ with a particular interest in those systems whose configuration space is the special Euclidean group $\mathsf {SE}(3) = \mathsf {SO}(3) \ltimes {\mathbb {R}}^{3}$. The key idea behind the work is the use of representations of ${\mathsf {G}} \ltimes V$ and their associated advected parameters. Specifically, we derive matching conditions for the modified potential exploiting the representations and advected parameters. Our motivating examples are a heavy top spinning on a movable base and an underwater vehicle with non-coincident centers of gravity and buoyancy. We consider a few different control problems for these systems, and show that our results give a general framework that reproduces our previous work on the former example and also those of Leonard on the latter. Also, in one of the latter cases, we demonstrate the advantage of our representation-based approach by giving a simpler and more succinct formulation of the problem.

Lie–Poisson brackets

While this paper focuses on the Lagrangian formulation of mechanical systems with broken symmetry, one can perform the Legendre transformation to obtain the Hamiltonian formulation of the systems as well. The main advantage of the Hamiltonian formulation is that it is more useful in finding the Casimirs.

1.1 Lie–Poisson bracket on ${\mathfrak {s}}^{*} = ({\mathfrak {g}} \ltimes V)^{*}$

Let ${\mathfrak {s}} = {\mathfrak {g}} \ltimes V$ be the Lie algebra of the semidirect product Lie group ${\mathsf {S}} \mathrel {\mathop :}={\mathsf {G}} \ltimes V$. The $(-)$-Lie–Poisson bracket on ${\mathfrak {s}}^{*}$ is given by (see Marsden et al. [22, 23])

$$\begin{aligned} \left\{ f,h\right\} _{{\mathfrak {s}}^{*}}(\mu ,p) = -{\left\langle \mu , \left[ \frac{\delta f}{\delta \mu }, \frac{\delta h}{\delta \mu } \right] \right\rangle } -{\left\langle p, \rho '{\left( \frac{\delta f}{\delta \mu }\right) } \frac{\partial h}{\partial p} - \rho '{\left( \frac{\delta h}{\delta \mu }\right) } \frac{\partial f}{\partial p} \right\rangle } \end{aligned}$$

(31)

We denote ${\mathfrak {s}}^{*}$ equipped with $\left\{ \,\cdot \,,\,\cdot \,\right\} _{{\mathfrak {s}}^{*}}$ by ${\mathfrak {s}}^{*}$.

Example 10

(Lie–Poisson bracket on $\mathfrak {se}(3)^{*}$) If ${\mathfrak {g}} = \mathfrak {so}(3)$ and $V = {\mathbb {R}}^{3}$, then ${\mathfrak {s}} = \mathfrak {se}(3)$. Using the expression for $\rho '$ from (4), (31) yields

$$\begin{aligned} \left\{ f,h\right\} _{\mathfrak {se}(3)^{*}}(\varvec{\Pi },{\mathbf {P}}) = -\varvec{\Pi }\cdot {\left( \frac{\partial f}{\partial \varvec{\Pi }} \times \frac{\partial h}{\partial \varvec{\Pi }} \right) } - {\mathbf {P}}\cdot {\left( \frac{\partial f}{\partial \varvec{\Pi }} \times \frac{\partial h}{\partial {\mathbf {P}}} - \frac{\partial h}{\partial \varvec{\Pi }} \times \frac{\partial f}{\partial {\mathbf {P}}} \right) }.\quad \end{aligned}$$

(32)

This is essentially the heavy top bracket upon replacing ${\mathbf {P}}$ by $\varvec{\Gamma }$. In our context, ${\mathbf {P}}$ stands for the linear impulse defined in (9), and so has a different physical meaning from $\varvec{\Gamma }$.

1.2 Lie–Poisson bracket on $({\mathfrak {s}} \ltimes X)^{*}$

We may describe those uncontrolled mechanical systems with broken symmetry shown in Sect. 3.3 as the Lie–Poisson equation on the dual $({\mathfrak {s}} \ltimes X)^{*}$ of the semidirect product Lie algebra ${\mathfrak {s}} \ltimes X$. Particularly, using the representation $\sigma$ defined in Sect. 3.2, the Lie–Poisson bracket on $({\mathfrak {s}} \ltimes X)^{*}$ is given by

$$\begin{aligned} \left\{ f,h\right\} _{({\mathfrak {s}} \ltimes X)^{*}}(\mu ,p,a) = \left\{ f,h\right\} _{{\mathfrak {s}}^{*}} -{\left\langle a, \sigma '{\left( \frac{\delta f}{\delta (\mu ,p)}\right) } \frac{\partial h}{\partial a} - \sigma '{\left( \frac{\delta h}{\delta (\mu ,p)}\right) } \frac{\partial f}{\partial a} \right\rangle }. \end{aligned}$$

(33)

Also, by considering a subrepresentation on $({\mathfrak {s}} \ltimes X)^{*}$, the controlled system (21) with potential shaping using the matching described in Sect. 4.2 may also be described in terms of the Lie–Poisson bracket on $({\mathfrak {s}} \ltimes {\tilde{X}})^{*}$.

Example 11

(Lie–Poisson bracket on $(\mathfrak {se}(3) \ltimes {\mathbb {R}}^{3})^{*}$) If ${\mathfrak {s}} = \mathfrak {se}(3)$ and $X = {\mathbb {R}}^{3}$, then, using the bracket (32) and also the expression for $\sigma '$ from (10), (33) gives

$$\begin{aligned} \left\{ f,h\right\} _{(\mathfrak {se}(3) \ltimes {\mathbb {R}}^{3})^{*}}(\varvec{\Pi },{\mathbf {P}},\varvec{\Gamma })&= -\varvec{\Pi }\cdot {\left( \frac{\partial f}{\partial \varvec{\Pi }} \times \frac{\partial h}{\partial \varvec{\Pi }} \right) } - {\mathbf {P}}\cdot {\left( \frac{\partial f}{\partial \varvec{\Pi }} \times \frac{\partial h}{\partial {\mathbf {P}}} \!-\! \frac{\partial h}{\partial \varvec{\Pi }} \times \frac{\partial f}{\partial {\mathbf {P}}} \right) } \\&\quad - \varvec{\Gamma }\cdot {\left( \frac{\partial f}{\partial \varvec{\Pi }} \times \frac{\partial h}{\partial \varvec{\Gamma }} - \frac{\partial h}{\partial \varvec{\Pi }} \times \frac{\partial f}{\partial \varvec{\Gamma }} \right) }. \end{aligned}$$

The uncontrolled underwater vehicle from Example 3 is governed by the Lie–Poisson equation with respect to this bracket. Note also that the heavy top on a movable base after potential shaping shown in Example 7 is also described in terms of the same bracket.

1.3 Lie–Poisson bracket on $({\mathfrak {s}} \ltimes (X \times Y))^{*}$

Matching described in Sect. 4.3 yields Lie–Poisson equation on the extended $({\mathfrak {s}} \ltimes (X \times Y))^{*}$ with the additional parameters living in $Y^{*}$. Using the representation $\tau$ defined in Sect. 4.3, we have the Lie–Poisson bracket on $({\mathfrak {s}} \ltimes (X \times Y))^{*}$ as follows:

$$\begin{aligned} \left\{ f,h\right\} _{({\mathfrak {s}} \ltimes (X \times Y))^{*}}(\mu ,p,a,b)&= \left\{ f,h\right\} _{{\mathfrak {s}}^{*}} \!-{\left\langle a,\!\right\rangle }{ \sigma '{\left( \frac{\delta f}{\delta (\mu ,p)}\right) } \frac{\partial h}{\partial a} \!- \sigma '{\left( \frac{\delta h}{\delta (\mu ,p)}\right) } \frac{\partial f}{\partial a} } \nonumber \\&\quad - {\left\langle b, \tau '{\left( \frac{\delta f}{\delta (\mu ,p)}\right) } \frac{\partial h}{\partial b} - \tau '{\left( \frac{\delta h}{\delta (\mu ,p)}\right) } \frac{\partial f}{\partial b} \right\rangle }.\quad \end{aligned}$$

(34)

Example 12

(Lie–Poisson bracket on $(\mathfrak {se}(3) \ltimes ({\mathbb {R}}^{3} \times ({\mathbb {R}}^{4} \times {\mathbb {R}}^{4})))^{*}$) Consider the case with ${\mathfrak {s}} = \mathfrak {se}(3)$, $X = {\mathbb {R}}^{3}$, and $Y = {\mathbb {R}}^{4} \times {\mathbb {R}}^{4}$. Using the expression for $\tau '$ from (18), (34) gives, using the shorthand $\Delta _{i} = (\varvec{\Delta }_{i},\delta _{i}) \in {\mathbb {R}}^{4}$ with $i = 1, 2$,

$$\begin{aligned}&\left\{ f,h\right\} _{(\mathfrak {se}(3) \ltimes ({\mathbb {R}}^{3} \times ({\mathbb {R}}^{4} \times {\mathbb {R}}^{4})))^{*}}(\varvec{\Pi }, {\mathbf {P}}, \varvec{\Gamma }, \Delta _{1}, \Delta _{2}) \\&\quad =-\varvec{\Pi }\cdot {\left( \frac{\partial f}{\partial \varvec{\Pi }} \times \frac{\partial h}{\partial \varvec{\Pi }} \right) } - {\mathbf {P}}\cdot {\left( \frac{\partial f}{\partial \varvec{\Pi }} \times \frac{\partial h}{\partial {\mathbf {P}}} - \frac{\partial h}{\partial \varvec{\Pi }} \times \frac{\partial f}{\partial {\mathbf {P}}} \right) }\\&\qquad - \varvec{\Gamma }\cdot {\left( \frac{\partial f}{\partial \varvec{\Pi }} \times \frac{\partial h}{\partial \varvec{\Gamma }} - \frac{\partial h}{\partial \varvec{\Pi }} \times \frac{\partial f}{\partial \varvec{\Gamma }} \right) } \\&\qquad - \sum _{i=1}^{2} \varvec{\Delta }_{i} \cdot {\left( \frac{\partial f}{\partial \varvec{\Pi }} \times \frac{\partial h}{\partial \varvec{\Delta }_{i}} - \frac{\partial h}{\partial \varvec{\Pi }} \times \frac{\partial f}{\partial \varvec{\Delta }_{i}} - \frac{\partial f}{\partial \delta _{i}} \frac{\partial h}{\partial {\mathbf {P}}} + \frac{\partial h}{\partial \delta _{i}} \frac{\partial f}{\partial {\mathbf {P}}} \right) } \\&\quad = \frac{\partial f}{\partial \varvec{\Pi }} \cdot {\left( \varvec{\Pi }\times \frac{\partial h}{\partial \varvec{\Pi }} + {\mathbf {P}}\times \frac{\partial h}{\partial {\mathbf {P}}} + \varvec{\Gamma }\times \frac{\partial h}{\partial \varvec{\Gamma }} + \sum _{i=1}^{2} \varvec{\Delta }_{i} \times \frac{\partial h}{\partial \varvec{\Delta }_{i}} \right) } \\&\qquad + \frac{\partial f}{\partial {\mathbf {P}}} \cdot {\left( {\mathbf {P}}\times \frac{\partial h}{\partial \varvec{\Pi }} - \sum _{i=1}^{2} \frac{\partial h}{\partial \delta _{i}} \varvec{\Delta }_{i} \right) } + \frac{\partial f}{\partial \varvec{\Gamma }} \cdot {\left( \varvec{\Gamma }\times \frac{\partial h}{\partial \varvec{\Pi }} \right) } \\&\qquad + \sum _{i=1}^{2} {\left( \frac{\partial f}{\partial \varvec{\Delta }_{i}} \cdot {\left( \varvec{\Delta }_{i} \times \frac{\partial h}{\partial \varvec{\Pi }} \right) } + \frac{\partial f}{\partial \delta _{i}} {\left( \varvec{\Delta }_{i} \cdot \frac{\partial h}{\partial {\mathbf {P}}} \right) } \right) }. \end{aligned}$$

This is the Lie–Poisson bracket for the controlled system (29) from Example 9. One sees from the expression that ${\mathbf {P}}\cdot (\varvec{\Delta }_{1} \times \varvec{\Delta }_{2}), {\left\| \varvec{\Gamma }\right\| }^{2}$, ${\left\| \varvec{\Delta }_{i}\right\| }^{2}$, $\varvec{\Gamma }\cdot \varvec{\Delta }_{i}$, $\varvec{\Delta }_{1} \cdot \varvec{\Delta }_{2}$ with $i = 1,2$ are Casimirs.