Jacobians

A Jacobian is a matrix that collects the partial derivatives of rates of change between two vectors. These rates of change are related by

$$\dot{a}=\mathbf{J}\dot{b}$$

$$\mathbf{J}= \left[\begin{matrix} j_{11} & j_{12} & \ldots & j_{1n} \\ j_{21} & j_{22} & \ldots & j_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ j_{m1} & j_{m2} & \ldots & j_{mn} \\ \end{matrix} \right]$$

where $j_{ij} = \frac{\partial \dot{a}_i}{\partial \dot{b}_j}$

Physical Meaning

$$\dot{q}_2 = J\dot{q}_1$$ $$\tau_1 = J\tau_2$$ $$J = \frac{r_1}{r_2}$$

Jacobian

$$\dot{x} = \mathbf{J} \dot{q}$$ $$\left[\begin{matrix}\dot{x}\\ \dot{y}\end{matrix}\right] = \mathbf{J} \left[\begin{matrix}\dot{q}_1\\ \dot{q}_2\end{matrix}\right]$$

Virtual Work

$$P_{in} = P_{out}$$ $$P_{in} = \tau^T \dot{q} = F^T \dot{y}=P_{out}$$ $$\tau^T \dot{q} = F^T \mathbf{J}\dot{q}$$ $$\tau^T = F^T \mathbf{J}$$ $$\tau = \mathbf{J}^T F$$

Torques and Forces

Forces applied at the end-effector can be related kinematically to torques felt by the motors using the Jacobian