Table 2 Definition of mathematical terms for the control-oriented model
Symbol | Description | Vector |
|---|---|---|
N | The number of links | |
l | The length of a link | |
m | Mass of each link | |
\(\phi _i\) | Normal direction distance between links i and \(i+1\) | \({\varvec{\phi }} \in \mathbb {R}^{N-1}\) |
\(\upsilon _{\phi ,i}\) | Relative velocity between links i and \(i+1\) | \({\varvec{\upsilon }}_{\phi } \in \mathbb {R}^{N-1}\) |
\(\theta\) | Orientation of the underwater snake robot | \(\theta \in \mathbb {R}\) |
\(\upsilon _{\theta }\) | Angular velocity of the underwater snake robot | \(\upsilon _{\theta } \in \mathbb {R}\) |
\((t_i, n_i)\) | Coordinates of the CM of link i in the \(t-n\) frame | \((\mathbf {t}, \mathbf {n}) \in \mathbb {R}^{2N}\) |
\((p_t, p_n)\) | Coordinates of the CM of the robot in the \(t-n\) frame | \((p_t, p_n) \in \mathbb {R}^2\) |
\((p_x, p_y)\) | Coordinates of the CM of the robot in the global frame | \((p_x, p_y) \in \mathbb {R}^2\) |
\((\upsilon _t, \upsilon _n)\) | Forward and normal direction velocity of the robot | \((\upsilon _t, \upsilon _n) \in \mathbb {R}^2\) |
\(u_{i}\) | Actuator force at joint i | \(\mathbf {u} \in \mathbb {R}^{N-1}\) |
\((f_{x,i}, f_{y,i})\) | Fluid force on link i in the global frame | \((\mathbf {f_{x}}, \mathbf {f_{y}}) \in \mathbb {R}^{2N}\) |
\((f_{t,i}, f_{n,i})\) | Fluid force on link i in the \(t-n\) frame | \((\mathbf {f_{t}}, \mathbf {f_{n}}) \in \mathbb {R}^{2N}\) |