Abstract
This chapter provides foundations not only for bilateral teleoperation but also for all of the subsequent chapters. Passivity, stability of dynamical systems, and several passivity-based motion control schemes are introduced.
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Notes
- 1.
Throughout this book, the notation \(\Vert x\Vert \) for a vector \(x\) describes the vector \(2\)-norm \(\Vert x\Vert = \sqrt{x^Tx}\) unless otherwise noted.
- 2.
See [159] for the details on the existence and uniqueness of the solution.
- 3.
A function \(S(x): {\mathbb R}^n \rightarrow {\mathbb R}\) is said to be positive definite if \(S(0) = 0\) and \(S(x) > 0\ {\forall x}\ne 0\). It is positive semidefinite if \(S(0) = 0\) and \(S(x) \ge 0\ {\forall x}\ne 0\) hold. Also, if \(-S\) is positive definite (semidefinite), then \(S\) is said to be negative definite (semidefinite).
- 4.
A map \(H: {\mathcal U} \rightarrow {\mathcal Y}\) is said to be causal if the output \((H(u))(\tau )\) at any time \(\tau \in {\mathbb R}_+\) is dependent only on the past and current profile of input \(u(t),\ t \le \tau \). See [318] for its formal definition.
- 5.
A matrix \(A \in {\mathbb R}^{n \times n}\) is said to be skew symmetric if \(A + A^T = 0\).
- 6.
The robot actually has one more degree of freedom.
- 7.
A matrix \(A \in {\mathbb R}^{n \times n}\) is said to be positive definite if it is symmetric (\(A = A^T\)) and \(x^T A x > 0 \ \forall x \ne 0\).
- 8.
The notation \(e \equiv 0\) for any signal \(e\) means \(e(t) = 0\ \ {\forall t}\in {\mathbb R}_+\).
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Hatanaka, T., Chopra, N., Fujita, M., Spong, M.W. (2015). Foundation: Passivity, Stability and Passivity-Based Motion Control. In: Passivity-Based Control and Estimation in Networked Robotics. Communications and Control Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-15171-7_2
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DOI: https://doi.org/10.1007/978-3-319-15171-7_2
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