Transformation properties of the Lagrange function

Ferrario, C.; Passerini, Arianna

doi:10.1590/s1806-11172008005000006

Cited by 1 publication

(8 citation statements)

References 17 publications

(24 reference statements)

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“…Continuing with the quasi-fully actuated form (42), compare Figure 10(top), we demonstrate how the global asymptotic stabilization problem, qu → 0 for t → ∞, where qu ≜ q u − q d denotes the output error, may be solved by adopting the classical energy-shaping and damping injection technique introduced in Reference 2. Our goal is to shape the potential energy of system Σ z such that the potential energy of the closed-loop system is radially unbounded and has a unique global minimum at the target point zd = [ q T d 0 ] T .…”

Section: Adopting the Energy-shaping Conceptmentioning

confidence: 90%

“…In Step 2 in Section 3.2, we introduced a set of state and input transforming equations, (39), and (41) to transform (37) into its QFA form (42). Note that this transformation does not represent a point transformation 41 or canonical transformation.…”

Section: Some Energy Considerationsmentioning

confidence: 99%

“…to (42), the potential function of system Σ z takes on the desired one,  d (z), see Reference 2 for details. The Hamiltonian of the controlled system is thus given by  d ( z, ̇z ) =  (q u , ̇z) +  d (z).…”

Section: Adopting the Energy-shaping Conceptmentioning

confidence: 99%

“…. We finish the transformation by premultiplying equation (A8) with T −T z to obtain the desired form (42).…”

Section: 𝝉( φ)mentioning

confidence: 99%

“…Note that this transformation does not represent a point transformation 41 or canonical transformation 37 . As a consequence, the principle of scalar invariance 42 does not hold, and, in general, the Hamiltonians associated with the systems

\sum_{z}

and

\sum_{\tilde{z}}

do not evaluate to the same values for

{\overline{u}}_{1} \neq 0

. We can, however, quantify the difference in energies between the open‐loop and closed‐loop systems

\begin{align} normalΔ ℋ & ≜ ℋ ((), {bold-italicT}_{z}^{prefix- 1} bold-italicz, {bold-italicT}_{z}^{prefix- 1} \overset{bold-italicz}{true}) prefix- ℋ ((), {bold-italicT}_{z}^{prefix- 1} \overset{bold-italicz}{true}, {bold-italicT}_{z}^{prefix- 1} \overset{\overset{bold-italicz}{true}}{true}) \\ = 𝒱_{e} false(bold-italicφ false) + \frac{1}{2} {((), \overset{bold-italicq}{true} + \overset{bold-italicφ}{true})}^{normalT} bold-italicB ((), \overset{bold-italicq}{true} + \overset{bold-italicφ}{true}) prefix- true[𝒱_{e} false(\overset{bold-italicφ}{true} false) + \frac{1}{2} {false(\overset{bold-italicq}{true} + \overset{\overset{bold-italicφ}{true}}{true} false)}^{normalT} bold-italicB false(\overset{bold-italicq}{true} + \overset{\overset{bold-italicφ}{true}}{true} false) true] . \end{align}

In case the closed‐loop motor inertia was shaped, one has to adapt the motor inertia matrix in the associated kinetic energy term accordingly.…”

Section: A Structure Preserving Input and State Transformationmentioning

confidence: 99%

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From underactuation to quasi‐full actuation: Aiming at a unifying control framework for articulated soft robots

Keppler

Ott

Albu‐Schäffer

2022

Intl J Robust & Nonlinear

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We establish a structure preserving state and input transformation that allows a class of underactuated Euler Lagrange systems to be treated as “quasi‐fully” actuated. In this equivalent quasi‐fully actuated form, the system is characterized by the same Lagrangian structure as the original one. This facilitates the design of control approaches that take into account the underlying physics of the system and that shape the system dynamics to a minimum extent. Due to smoothness constraints on the new input vector that acts directly on the noncollocated coordinates, we coin the term quasi‐fully actuated. The class of Euler–Lagrange systems we consider is the class of articulated soft robots with nonlinear spring characteristics that are modeled with a block diagonal inertia matrix. We illustrate how the quasi‐fully actuated form enables the direct transfer of control concepts that have been derived for fully actuated manipulators. We adopt the popular energy‐shaping and two passivity‐based concepts. The exemplary adoptions of the PD+ and Slotine and Li controllers allow us to solve the task‐space tracking problem for highly elastic joint robots with nonlinear spring characteristics. These control schemes allow compliant behavior of the robot's TCP to be specified with respect to a reference trajectory. A key aspect of the presented framework is that it enables the adoption of rigid joint controllers as well as concepts underlying the original stability analysis. We believe that our framework presents an important step toward unifying the control design for rigid and articulated soft robots.

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Section: Adopting the Energy-shaping Conceptmentioning

confidence: 90%