Multibody dynamic modeling and motion analysis of flexible robot considering contact

Abstract

The purpose of this research is to present an alternative multibody dynamic model for soft robots and to analyze the intrinsic mechanism of motion. It is difficult to directly apply traditional robot modeling methods due to the large structural deformation of soft walking robots. This paper establishes the dynamic modeling of a soft robot system with contact/impact based on the corotational formulation of the special Euclidean group $SE$(2). The experiments are designed to verify the dynamic model of the robot. The history of the marked points on the robot prototype is measured in real time by an ARAMIS Adjustable Camera System. Based on the dynamic model, we conducted an in-depth analysis of the entire process through which the robot achieves directional walking utilizing complex friction characteristics. Notably, the robot’s kick-up phenomenon attracted our attention, and an analytical model for predicting the critical drive acceleration is proposed. The conditions and mechanisms of the robot’s kick-up are analyzed, and effective direction is provided for designing new drive laws. Finally, several sets of key parameters affecting the walking efficiency are analyzed using the multibody model, which can provide scientific guidance for the material selection and optimization of the robot. The presented dynamic modeling approach can be freely extended to other soft robots, which will provide valuable references for the design and analysis of soft robots.

Appendix

1.1 A.1 Lie group

The Lie group (Group) is a collection of operations specifying some kind. A Lie group is a smooth differential manifold with group structure. The group action is compatible with the differential structure. The dynamics of a flexible multibody system can be described by a k-dimensional manifold $G$ with a Lie group structure. The product and inverse operations of the Lie group $G$ are smooth maps [39]. Lie groups follow the following axioms:

(i) closure: $\mathbf{q}_{1} \cdot \mathbf{q}_{2} \in g$,

(ii) identity: $\mathbf{e} \cdot \mathbf{q}_{1} = \mathbf{q}_{1} \cdot \mathbf{e} = \mathbf{q}_{1}$,

(iii) invertibility: $\mathbf{q}^{ - 1} \cdot \mathbf{q} = \mathbf{q} \cdot \mathbf{q}^{ - 1} = \mathbf{e}$,

(iv) associativity: $(\mathbf{q}_{1} \cdot \mathbf{q}_{2}) \cdot \mathbf{q}_{3} = \mathbf{q}_{1} \cdot (\mathbf{q}_{2} \cdot \mathbf{q}_{3})$,

where the special element $\mathbf{e}$ is the identity element in this group. The Cartesian vector space $\mathbb{R}^{n}$, the special orthogonal group $SO(n)$, and the special Euclidean group $SE(n)$ all belong to the Lie group.

1.2 A.2 Lie algebra and left translation

The identity element $\mathbf{e}$ is such that $\mathbf{e} \cdot \mathbf{q} = \mathbf{q} \cdot \mathbf{e} = \mathbf{q}, \forall \mathbf{q} \in G$. $T_{q}G$ denotes the tangent space at the point $\mathbf{q} \in G$. The Lie algebra is defined as the tangent space at the identity $g = T_{e}G$. The Lie algebra is isomorphic to the Cartesian vector space and can be linearly mapped by the operator ($\boldsymbol{\cdot}$):

$$ (\tilde{ \cdot} ):\mathbb{R}^{k} \to g, \mathbf{v} \mapsto \tilde{\mathbf{v}}. $$

(61)

Since group operations generally do not satisfy the exchange law, they can be divided into left and right translations according to the position of the action of the translation. For the element $\mathbf{y} \in G$ the translation is defined as

$$ L_{q}(\mathbf{y}):G \to G, \mathbf{y} \mapsto \mathbf{q} \cdot \mathbf{y}, $$

(62)

$$ R_{\mathbf{q}}(\mathbf{y}):G \to G, \mathbf{y} \mapsto \mathbf{y} \cdot \mathbf{q}. $$

(63)

$L_{q}(\mathbf{y})$ and $R_{\mathbf{q}}(\mathbf{y})$ are the left and right translation, respectively. This paper mainly utilizes the left translation property.

Let the derivative of the group element $\mathbf{y}$ with respect to $t$ be defined as $d_{t}(\mathbf{y})$:

$$ d_{t}(\mathbf{q} \cdot \mathbf{y}) = d_{t}\left ( L_{\mathbf{q}}(\mathbf{y}) \right ) = DL_{\mathbf{q}}(\mathbf{y}) = \mathbf{q} \cdot d_{t}(\mathbf{y}), $$

(64)

where $d_{t}(\mathbf{y}) \in T_{\mathbf{y}}G$ is the tangent space of the manifold at element $\mathbf{y}$. As shown in Fig. 6, $DL_{\mathbf{q}}(\mathbf{y})$ defines the translation between two tangent spaces:

$$ DL_{\mathbf{q}}(\mathbf{y}):T_{\mathbf{y}}G \to T_{\mathbf{q}}G. $$

(65)

At the identity element $\mathbf{e} \in G$, $DL_{\mathbf{q}}(\mathbf{e})$ denotes the mapping of the tangent space at the identity element to an arbitrary point $\mathbf{q}$. Thus, a linear mapping for $\tilde{\mathbf{v}} \in g$ into any tangent space $T_{\mathbf{q}}G$ is defined as

$$ DL_{\mathbf{q}}(\mathbf{e}):g \to T_{\mathbf{q}}G, \tilde{\mathbf{v}} \mapsto DL_{\mathbf{q}}(\mathbf{e}) \cdot \tilde{\mathbf{v}}. $$

(66)

Thereby, a connection between Lie groups and Lie algebras with respect to time derivatives is established. The derivative of a group element $\mathbf{q}$ can be expressed as

$$ \dot{\mathbf{q}} = DL_{\mathbf{q}}(\mathbf{e}) \cdot \tilde{\mathbf{v}}. $$

(67)

1.3 A.3 Exponential map

The exponential mapping builds a bridge between Lie groups and Lie algebras. We take the left invariant derivative as an example:

$$ \mathbf{q}(t) = \mathbf{q}_{0}\exp (\tilde{\mathbf{v}}t), $$

(68)

where $\mathbf{q}_{0}$ is the constant of integration, which can be interpreted as the value of the element at the initial moment. The operator exp can be expressed in the form of a series expansion. Let $\tilde{\mathbf{v}}t = \tilde{\boldsymbol{\tau}}$. It follows that

$$ \exp (\tilde{\boldsymbol{\tau}} )\mathop{ =} \limits ^{\mathrm{def}}\sum _{i = 0}^{\infty} \frac{1}{i!} \tilde{\boldsymbol{\tau}}^{i}. $$

(69)

When $\mathbf{q}_{0}$ is the identity element $\mathbf{e}$, there is the following exponential mapping relation:

$$ \exp :g \to G, \tilde{\boldsymbol{\tau}} \to \mathbf{q} = \exp (\tilde{\boldsymbol{\tau}} ). $$

(70)

Equation (68) shows that any point on the manifold can be obtained from the tangent vector at the identity element through exponential mapping. The linearized exponential mapping is obtained by the derivation of Eq. (67). We have

$$ \mathbf{T}\left ( \tilde{\boldsymbol{\tau}} \right )\mathop{ =} \limits ^{\mathrm{def}}\frac{D\exp (\tilde{\boldsymbol{\tau}} )}{D\boldsymbol{\tau}} = \sum _{i = 0}^{\infty} \frac{( - 1)^{i}}{(i + 1)!} \hat{\boldsymbol{\tau}}^{i}. $$

(71)

$\mathbf{T}$ is the tangent operator of an exponential map, which maps the variation of the Lie algebra at the identity element to the local tangent space of any point. The left invariant derivative is as follows:

$$ \exp (\widetilde{\boldsymbol{\tau} + \delta \boldsymbol{\tau}} ) \approx \exp (\tilde{\boldsymbol{\tau}} )\exp (\widetilde{\mathbf{T}\delta \boldsymbol{\tau}} ). $$

(72)