Table 1 Definition of mathematical terms for the complex model
Symbol | Description | Vector |
|---|---|---|
n | The number of links | |
l | The half length of a link | |
m | Mass of each link | |
J | Moment of inertia of each link | |
\(\theta _{i}\) | Angle between link i and the global x axis | \({\varvec{\theta }}\) \(\in \mathbb {R}^n\) |
\(\phi _{i}\) | Angle of joint i | \({\varvec{\phi }}\) \(\in \mathbb {R}^{n-1}\) |
\((x_i, y_i)\) | Global coordinates of the CM of link i | \(\mathbf {X},\mathbf {Y}\) \(\in \mathbb {R}^n\) |
\((p_x, p_y)\) | Global coordinates of the CM of the robot | \(\mathbf {p}_\text {{CM}}\) \(\in \mathbb {R}^2\) |
\(u_{i}\) | Actuator torque of joint between link i and link \(i+1\) | \(\mathbf {u}\) \(\in \mathbb {R}^{n-1}\) |
\(u_{i-1}\) | Actuator torque of joint between link i and link \(i-1\) | \(\mathbf {u}\) \(\in \mathbb {R}^{n-1}\) |
\(f_{x,i}\) | Fluid force on link i in x direction | \({\mathbf {f_{x}}}\) \(\in \mathbb {R}^n\) |
\(f_{y,i}\) | Fluid force on link i in y direction | \({\mathbf {f_{y}}}\) \(\in \mathbb {R}^n\) |
\(\tau _i\) | Fluid torque on link i | \({\varvec{\tau }} \in \mathbb {R}^n\) |
\(h_{x,i}\) | Joint constraint force in x direction on link i from link \(i+1\) | \({\mathbf {h_{x}}}\) \(\in \mathbb {R}^{n-1}\) |
\(h_{y,i}\) | Joint constraint force in y direction on link i from link \(i+1\) | \({\mathbf {h_{y}}}\) \(\in \mathbb {R}^{n-1}\) |
\(h_{x,i-1}\) | Joint constraint force in x direction on link i from link \(i-1\) | \({\mathbf {h_{x}}}\) \(\in \mathbb {R}^{n-1}\) |
\(h_{y,i-1}\) | Joint constraint force in y direction on link i from link \(i-1\) | \({\mathbf {h_{y}}}\) \(\in \mathbb {R}^{n-1}\) |