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Motion analysis of two-link nonholonomic swimmers

Abstract

This paper presents a tool for analyzing the motion of two-link nonholonomic swimmers. We refer to these systems as Land-sharks, which are a generalization of the well known Roller Racers. By exploiting the symmetry of the system, we are able to reduce the equations of motion and construct the scaled momentum evolution equation. This unveils a very useful and intuitive Land-shark motion analysis tool based on the partitioning of the mass and geometry parameter space. In particular, this partitioning reveals that, as opposed to the Roller Racer, the Land-shark’s momentum can be increased and decreased, i.e., the system can be stopped. This is done through the use of steering, which is the system’s only input. Furthermore, we explore the problem of modeling frictional slip by assessing the applicability of a previously proposed friction model to the oscillatory locomotion of the Land-shark. Results show that the proposed friction model is generally applicable to two-link nonholonomic mechanical systems, which is an important step toward establishing the generality of the friction model for nonholonomic mechanical systems.

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Acknowledgements

This work is supported by the Lebanese National Council for Scientific Research (LNCSR), the University Research Board (URB) of the American University of Beirut, and the Munib and Angela Masri Institute. The authors would like to thank the reviewers for their insightful comments, which helped in improving the work presented in this paper.

Author information

Correspondence to Salah Bazzi.

Electronic supplementary material

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Supplementary material 1 (mp4 3494 KB)

Supplementary material 1 (mp4 3494 KB)

Appendices

Appendix

A Proofs of Lemmas

A.1 Proof of Lemma 1

Proof

$$\begin{aligned}&\underline{\text {For} \quad \frac{M_2}{M_1}>1}\\&\frac{L_1^2 M_1-L_2^2 M_2}{L_1 L_2 \left( M_2-M_1\right) }<-1\\&L_1^2 M_1-L_2^2 M_2<L_1 L_2 \left( M_1-M_2\right) \\&L_1^2 M_1-L_1 L_2 M_1<L_2^2 M_2-L_1 L_2 M_2\\&\left( L_1^2-L_1 L_2\right) M_1<\left( L_2^2-L_1 L_2\right) M_2\\&\left( \frac{L_1}{L_2}-1\right) M_1<\left( \frac{L_2}{L_1}-1\right) M_2\\&\text {If} \quad \quad \frac{L_2}{L_1}>1\\&\implies \frac{M_2}{M_1}>-\frac{L_1}{L_2} \end{aligned}$$

(always holds since masses and lengths are positive).

$$\begin{aligned}&\text {If} \quad \quad \frac{L_2}{L_1}<1\\&\quad \implies \frac{M_2}{M_1}<-\frac{L_1}{L_2} \end{aligned}$$

(never holds since masses and lengths are positive).

$$\begin{aligned}&\underline{\text {For} \quad \frac{M_2}{M_1}<1}\\&\frac{L_1^2 M_1-L_2^2 M_2}{L_1 L_2 \left( M_2-M_1\right) }>1\\&L_1^2 M_1-L_2^2 M_2<L_1 L_2 \left( M_2-M_1\right) \\&\left( L_1^2+L_1 L_2\right) M_1<\left( L_2^2+L_1 L_2\right) M_2\\&\left( \frac{L_1}{L_2}+1\right) M_1<\left( \frac{L_2}{L_1}+1\right) M_2\\&\implies \frac{M_2}{M_1}>\frac{L_1}{L_2}. \end{aligned}$$

\(\square \)

A.2 Proof of Lemma 2

Proof

$$\begin{aligned}&\underline{\text {For} \quad \frac{M_2}{M_1}>1}\\&\frac{L_1^2 M_1-L_2^2 M_2}{L_1 L_2 \left( M_2-M_1\right) }>1\\&L_1^2 M_1-L_2^2 M_2>L_1 L_2 \left( M_2-M_1\right) \\&\left( L_1^2+L_1 L_2\right) M_1>\left( L_2^2+L_1 L_2\right) M_2\\&\left( \frac{L_1}{L_2}+1\right) M_1>\left( \frac{L_2}{L_1}+1\right) M_2\\&\implies \frac{M_2}{M_1}<\frac{L_1}{L_2}\\&\underline{\text {For} \quad \frac{M_2}{M_1}<1}\\&\frac{L_1^2 M_1-L_2^2 M_2}{L_1 L_2 \left( M_2-M_1\right) }<-1\\&L_1^2 M_1-L_2^2 M_2>L_1 L_2 \left( M_1-M_2\right) \\&\left( L_1^2-L_1 L_2\right) M_1>\left( L_2^2-L_1 L_2\right) M_2\\&\left( \frac{L_1}{L_2}-1\right) M_1>\left( \frac{L_2}{L_1}-1\right) M_2\\&\text {If} \quad \quad \frac{L_2}{L_1}>1\\&\implies \frac{M_2}{M_1}<-\frac{L_1}{L_2}\\ \end{aligned}$$

(never holds since masses and lengths are positive).

$$\begin{aligned}&\text {If} \quad \quad \frac{L_2}{L_1}<1\\&\implies \frac{M_2}{M_1}>-\frac{L_1}{L_2}\\ \end{aligned}$$

(always holds since masses and lengths are positive). \(\square \)

A.3 Proof of Lemma 3

Proof

$$\begin{aligned}&\underline{\text {For} \quad \frac{M_2}{M_1}>1}\\&\frac{L_1^2 M_1-L_2^2 M_2}{L_1 L_2 \left( M_2-M_1\right) }<1\\&L_1^2 M_1-L_2^2 M_2<L_1 L_2 \left( M_2-M_1\right) \\&\left( \frac{L_1}{L_2}+1\right) M_1<\left( \frac{L_2}{L_1}+1\right) M_2\\&\implies \frac{M_2}{M_1}>\frac{L_1}{L_2}\\&\frac{L_1^2 M_1-L_2^2 M_2}{L_1 L_2 \left( M_2-M_1\right) }>-1\\&L_1^2 M_1-L_2^2 M_2>L_1 L_2 \left( M_1-M_2\right) \\&\left( \frac{L_1}{L_2}-1\right) M_1>\left( \frac{L_2}{L_1}-1\right) M_2\\&\text {If} \quad \quad \frac{L_2}{L_1}>1\\&\implies \frac{M_2}{M_1}<-\frac{L_1}{L_2} \end{aligned}$$

(never holds since masses and lengths are positive).

$$\begin{aligned}&\text {If} \quad \quad \frac{L_2}{L_1}<1\\&\implies \frac{M_2}{M_1}>-\frac{L_1}{L_2} \end{aligned}$$

(always holds since masses and lengths are positive).

$$\begin{aligned}&\underline{\text {For} \quad \frac{M_2}{M_1}<1}\\&\frac{L_1^2 M_1-L_2^2 M_2}{L_1 L_2 \left( M_2-M_1\right) }<1\\&L_1^2 M_1-L_2^2 M_2>L_1 L_2 \left( M_2-M_1\right) \\&\left( \frac{L_1}{L_2}+1\right) M_1>\left( \frac{L_2}{L_1}+1\right) M_2\\&\implies \frac{M_2}{M_1}<\frac{L_1}{L_2}\\&\frac{L_1^2 M_1-L_2^2 M_2}{L_1 L_2 \left( M_2-M_1\right) }>-1\\&L_1^2 M_1-L_2^2 M_2<L_1 L_2 \left( M_1-M_2\right) \\&\left( \frac{L_1}{L_2}-1\right) M_1<\left( \frac{L_2}{L_1}-1\right) M_2\\&\text {If} \quad \quad \frac{L_2}{L_1}>1\\&\implies \frac{M_2}{M_1}>-\frac{L_1}{L_2} \end{aligned}$$

(always holds since masses and lengths are positive).

$$\begin{aligned}&\text {If} \quad \quad \frac{L_2}{L_1}<1\\&\implies \frac{M_2}{M_1}<-\frac{L_1}{L_2} \end{aligned}$$

(never holds since masses and lengths are positive). \(\square \)

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Bazzi, S., Shammas, E., Asmar, D. et al. Motion analysis of two-link nonholonomic swimmers. Nonlinear Dyn 89, 2739–2751 (2017) doi:10.1007/s11071-017-3622-y

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Keywords

  • Geometric mechanics
  • Nonholonomic motion planning
  • Frictional slip
  • Robotic locomotion