Update paper.md

paper.md

@@ -58,13 +58,13 @@ $$\begin{array}{cc}
        & \boldsymbol{\tau}_m \leq \boldsymbol{\tau} \leq \boldsymbol{\tau}_M \\
 \end{array}$$
 
-where $\mathbf{w}_i$ are the task weights for the i-th task, $\mathbf{J}_i$ is the respective robot Jacobian, $\dot{\mathbf{v}}_{d,i}$ the desired task space acceleration,  $\mathbf{q},\dot{\mathbf{q}},\ddot{\mathbf{q}}$ the joint positions, velocities, and accelerations $\mathbf{H}$ the mass-inertia matrix, $\mathbf{h}$ the vector of gravity and Coriolis forces, $\boldsymbol{\tau}$ the robot joint torques, $\mathbf{f}$ the contact wrenches,  $\mathbf{J}_c$ the contact Jacobian,  and $\boldsymbol{\tau}_m,\boldsymbol{\tau}_M$ the joint torque limits. To implement a simple Cartesian position controller for, e.g., controlling the end effector of a robot manipulator, the following PD-controller can be used to generate $\dot{\mathbf{v}}_d$:
+where $\mathbf{w}_i$ are the task weights for the i-th task, $\mathbf{J}_i$ is the respective robot Jacobian, $\dot{\mathbf{v}}_{d,i}$ the desired task space acceleration,  $\mathbf{q},\dot{\mathbf{q}},\ddot{\mathbf{q}}$ the joint positions, velocities, and accelerations, $\mathbf{H}$ the mass-inertia matrix, $\mathbf{h}$ the vector of gravity and Coriolis forces, $\boldsymbol{\tau}$ the robot joint torques, $\mathbf{f}$ the contact wrenches,  $\mathbf{J}_c$ the contact Jacobian,  and $\boldsymbol{\tau}_m,\boldsymbol{\tau}_M$ the joint torque limits. To implement a simple Cartesian position controller for, e.g., controlling the end effector of a robot manipulator, the following PD-controller can be used to generate $\dot{\mathbf{v}}_d$:
 
 $$
 \dot{\mathbf{v}}_d = \dot{\mathbf{v}}_r + \mathbf{K}_d(\mathbf{v}_r-\mathbf{v}) + \mathbf{K}_p(\mathbf{x}_r-\mathbf{x})
 $$
 
-where $\mathbf{K}_p,\mathbf{K}_d$ are gain matrices, $\mathbf{x},\mathbf{v}$ the end effector position and velocity, $\dot{\mathbf{v}}_r,\mathbf{v}_r,\mathbf{x}_r$ the reference acceleration, velocity, and position. Figure \ref{fig:snippet} shows a C++ code snippet from ARC-OPT, which implements the above QP on a KUKA iiwa robot arm (no contacts, fixed base robot). In the code example, first the robot model is set up using the robot URDF, then the QP-solver (qpOASES), the WBC scene and a Cartesian controller are configured. In the subsequent control loop, a circular trajectory is tracked in Cartesian space. The full example can be found in the ARC-OPT tutorials\footnote{\url{https://github.com/ARC-OPT/wbc/blob/master/tutorials/kuka_iiwa/cart_pos_ctrl_dynamic.cpp}}. Figure \ref{fig:screenshots} shows a visualization of the resulting robot motion. While the example implements a simple Cartesian controller, more complex problems can be composed using multiple controllers to track different targets, e.g., leg positions and center of mass for a walking robot.
+where $\mathbf{K}_p,\mathbf{K}_d$ are gain matrices, $\mathbf{x},\mathbf{v}$ the end effector position and velocity, $\dot{\mathbf{v}}_r,\mathbf{v}_r,\mathbf{x}_r$ the reference acceleration, velocity, and position. Figure \ref{fig:snippet} shows a C++ code snippet from ARC-OPT, which implements the above QP on a KUKA iiwa robot arm (no contacts, fixed base robot). In the code example, at first the robot model is set up using the URDF file, then the QP-solver (qpOASES), the WBC scene and a Cartesian controller are configured. In the subsequent control loop, a circular trajectory is tracked in Cartesian space. The full example can be found in the ARC-OPT tutorials\footnote{\url{https://github.com/ARC-OPT/wbc/blob/master/tutorials/kuka_iiwa/cart_pos_ctrl_dynamic.cpp}}. Figure \ref{fig:screenshots} shows a visualization of the resulting robot motion. While the example implements a simple Cartesian controller, more complex problems can be composed using multiple controllers to track different targets, e.g., leg positions and center of mass for a walking robot.
 
 ![Minimal code example for Cartesian position control on a KUKA iiwa robot\label{fig:snippet}](snippet.png)