In the previous post, we show how the equations of motion of an \(n\)-DOF open-chain torque-actuated robot manipulator can be derived: \[ \mathbf{M(q)\ddot{q}+C(q,\dot{q})\dot{q}+g(q)}=\boldsymbol{\tau}_\text{in} \tag{1} \] The notations of the equation are self-explanatory, and we assume there are no external forces applied to the robot. We introduce the Operational Space Control method first proposed by Khatib [1].
We consider the task variable \(\mathbf{x}\in\mathbb{R}^{m}\), which can be expressed as a function of \(\mathbf{q}\), i.e., \(\mathbf{x=x(q)}\). Mostly, \(\mathbf{x}\) is the 3D Cartesian position of the end-effector.
Given the Jacobian matrix \(\mathbf{J(q)}\), The Operational Space
Control defines the torque input \(\boldsymbol{\tau}_{in}\equiv\mathbf{J(q)^TF_{in}}\),
where given \(\mathbf{F_{in}}\in\mathbb{R}^{m}\) the
torque input is well-defined. Moreover, the time derivatives of the task
variable \(\mathbf{x(q)}\) provides us:
\[
\mathbf{\dot{x}(q)=J(q)\dot{q}}
~~~~~~~~~\mathbf{\ddot{x}(q)=\dot{J}(q)\dot{q} + J(q)\ddot{q}} \tag{2}
\] Since \(\mathbf{M(q)}\) is always
invertible for all \(\mathbf{q}\),
Eq. (1) and (2) can be reformulated as: \[
\mathbf{\ddot{q}=M^{-1}(q) \{\boldsymbol{\tau}_\text{in}
-C(q,\dot{q})\dot{q}+g(q)} \} ~~~~~~
\mathbf{\ddot{x}(q)}=\mathbf{\dot{J}(q)\dot{q}+J(q) \big\{ M^{-1}(q) [
J(q)^TF_{in} -C(q,\dot{q})\dot{q}+g(q) ] \big\} }
\] We denote \(\mathbf{J(q)M^{-1}(q)J^T(q)}\equiv
\boldsymbol{\Lambda}^{-1}(\mathbf{q})\).