Input Decoupling of Lagrangian Systems via Coordinate Transformation: General Characterization and its Application to Soft Robotics

TL;DR

This work tackles the Input Decoupling (ID) problem for Lagrangian systems by introducing actuation coordinates that render the input work directly on a subset of configuration variables, yielding a collocated form with $\tau_{\theta} = u$. It develops necessary and sufficient conditions for the existence of such coordinates across fully actuated, overactuated, and underactuated dynamics, anchored in power invariance and an integrability condition $J_g(q) = A^T(q)$. A key result is that thread-like actuators inherently admit actuation coordinates, extending to continuum soft robots via Geometric Variable Strain modeling, and enabling decoupled control strategies. The theory is leveraged to extend damping-based regulators to collocated underactuated systems, with demonstrations on tendon-driven soft robots showing reduced steady-state errors when controlling in the actuation coordinates. Overall, the paper provides a principled, coordinate-based pathway to simplify and robustify control of complex Lagrangian systems, especially in soft robotics, by exploiting actuation coordinates and power-invariant transformations.

Abstract

Suitable representations of dynamical systems can simplify their analysis and control. On this line of thought, this paper aims to answer the following question: Can a transformation of the generalized coordinates under which the actuators directly perform work on a subset of the configuration variables be found? Not only we show that the answer to this question is yes, but we also provide necessary and sufficient conditions. More specifically, we look for a representation of the configuration space such that the right-hand side of the dynamics in Euler-Lagrange form becomes $[\boldsymbol{I} \; \boldsymbol{O}]^{T}\boldsymbol{u}$, being $u$ the system input. We identify a class of systems, called collocated, for which this problem is solvable. Under mild conditions on the input matrix, a simple test is presented to verify whether a system is collocated or not. By exploiting power invariance, we provide necessary and sufficient conditions that a change of coordinates decouples the input channels if and only if the dynamics is collocated. In addition, we use the collocated form to derive novel controllers for damped underactuated mechanical systems. To demonstrate the theoretical findings, we consider several Lagrangian systems with a focus on continuum soft robots.

Figures (8)

• Figure 1: Graphical representation of the proposed change of coordinates addressing the input decoupling problem for a fully actuated Lagrangian system. In the ${\boldsymbol{q}}$ coordinates, the input $\boldsymbol{u}$ undergoes a nonlinear transformation through the actuation matrix $\boldsymbol{A}^{{}\!}_{{}}({\boldsymbol{q}})$ when performing work on ${\boldsymbol{q}}$, i.e., $\boldsymbol{\tau}_{{\boldsymbol{q}}} = \boldsymbol{A}^{{}\!}_{{}}({\boldsymbol{q}})\boldsymbol{u}$. The proposed change of coordinates $\boldsymbol{\theta} = \boldsymbol{h}({\boldsymbol{q}})$ bends the configuration space so that each component of $\boldsymbol{u}$ acts directly on one component of $\boldsymbol{\theta}$, namely $\boldsymbol{\tau}_{\boldsymbol{\theta}} = \boldsymbol{u}$. The existence of this transformation is possible because of the conservation of power $\dot{\mathcal{H}}_{{\boldsymbol{q}}}$, represented by the yellow area, under change of coordinates, i.e., $\dot{\mathcal{H}}_{{\boldsymbol{q}}}=\dot{\mathcal{H}}_{\boldsymbol{\theta}}$.
• Figure 2: A geostationary satellite actuated by a normal force $u_1$ and a tangential force $u_2$. The body configuration is described by the distance $q_1$ from the Earth center and the angle $q_2$ with respect to the horizontal axis. Only the normal force is collocated because it performs work directly and only on $q_1$.
• Figure 3: A planar mechanism with two rotational joints having configuration ${\boldsymbol{q}} \in \mathbb{S}^{1} \times \mathbb{S}^{1}$. Positive (relative) rotations are counted counterclockwise. For $i = 1, 2$, the force $u_{i}$ actuates a cart coupled to the robot by a linear spring of stiffness $k_i$. The dynamics of the carts is negligible, and thus the forces $\boldsymbol{u}$ act instantaneously on the mechanism.
• Figure 4: A tendon-driven rotational joint having a single configuration variable $q \in \mathbb{S}^{1}$. The cables tension $u_1$ and $u_2$ generate a torque at the joint.
• Figure 5: A continuum soft robot discretized into two bodies. Under the piecewise constant curvature assumption, the configuration of each body is described by its curvature $\kappa_{i}$, bending angle $\phi_{i}$ and elongation $\delta L_{i}$ for $i = 1, 2$. Three tendons run from the base to the tip, and spatial motion is obtained by applying a suitable cable tension $\boldsymbol{u} \in \mathbb{R}^{3}$.

Theorems & Definitions (11)

• Remark
• Remark
• Theorem 1
• proof
• Corollary 1
• proof
• Theorem 2
• proof
• Remark
• Corollary 2