Magnetic Levitation for Soft-Tethered Capsule Colonoscopy Actuated With a Single Permanent Magnet

一种动态控制方法，用于胶囊重力抵消，使胶囊悬浮起来。比在结肠模型里硬拖着走快多了。

磁悬浮法用于由单个永磁体驱动的软尾绳胶囊结肠镜检查：一种动态控制方法
Magnetic Levitation for Soft-Tethered Capsule Colonoscopy Actuated With a Single Permanent Magnet：A Dynamic Control Approach [1]
Paper Link
Authors: Pittiglio, Giovanni, et al.
2019, IEEE robotics and automation letters (RA-L)

0. 摘要 Abstract

我们提出一种控制策略，这个策略用以抵消重力，获得胶囊的悬浮。这个技术，基于一个非线性反推方法，能够限制与结肠壁的接触，减少阻力，避免与内褶皱的接触，并促进非平面腔的检查。这个方法在一个实验装置中被验证，体现面向结肠镜检查的一个通用场景。实验展示了我们能达到与结肠壁19.5%的接触率，与之前提出方法大约100%相比。另外，我们展示这个控制方法能被用来导航胶囊穿过一个更真实的环境，一个结肠仿体，在有竞争力的时间内。
We propose a control strategy which, counteracting gravity, achieves levitation of the capsule. This technique, based on a nonlinear backstepping approach, is able to limit contact with the colon walls, reducing friction, avoiding contact with internal folds, and facilitating the inspection of nonplanar cavities. The approach is validated on an experimental setup, which embodies a general scenario faced in colonoscopy. The experiments show that we can attain 19.5% of contact with the colon wall, compared to the almost 100% of previously proposed approaches. Moreover, we show that the control can be used to navigate the capsule through a more realistic environment—a colon phantom—with reasonable completion time

1. 介绍 Introduction

然而，这个平台的一个可能的限制是胶囊到EPM的持续的吸引，并且缺少重力补偿。这可能导致胶囊在结肠的解剖学上的复杂且非结构化的环境中困住，并且可能阻碍通过一个陡峭斜坡的运动。
However, a potential limitation of this platform is the continuous attraction of the capsule to the EPM and lack of gravity compensation. This may cause the capsule to become trapped in the anatomically complex and unstructured environment of the colon and may hinder locomotion through a steeply sloping lumen.

这个控制策略能够带来很大的好处当它促进避障，接触力的下降，同时因此，降低阻力和外伤危险或不舒适。它也可能辅助导航在结肠的倾斜区域。
This control strategy can bring significant benefit as it facilitates the avoidance of obstacles (eg. tissue folds), a reduction in contact force and therefore, a reduction in both friction and risk of trauma or discomfort. It may also assist with navigating sloped regions of the colon.

2. 理论 Method

更进一步的是，当考虑仅仅单个磁源，磁场和它的梯度的点控制不像在使用多线圈时那么直接。
Moreover, when considering only a single magnetic source, point-wise control of the magnetic field and its gradient is not as straightforward as in using multiple coils.

整体控制策略基于反推技术，并且全局稳定性通过李雅普诺夫方法的方式被正式证明。这在假设下被保证，假设是IPM的期望轨迹是一个时间的分段函数。这意味着IPM的期望速度和加速度能被忽略。在这样的条件下，一个PD控制器能被设计来调整IPM并获得渐进收敛性。做出的假设不干扰控制器的设计，也不在任何情况下有限制，当一个平滑的规划被获得的时候。
The overall control strategy is based on the backstepping technique and the global stability is formally proved by means of a Lyapunov-based approach. This is guaranteed under the assumption that the desired trajectory of the IPM is a piecewiseconstant function of the time. This means that desired velocity and acceleration of the IPM can be neglected. In this condition, a PD controller can be designed to steer the IPM and achieve asymptotic convergence. The assumption made does not interfere with the design of the controller, nor is limiting in any case when a smooth planning can be achieved.

3. 动态控制 Dynamic Control

我们考虑一种用于两个环的反推方法，位姿环和力环。后者，被考虑为一个内环，被设计来保证作用在IPM的实际力到期望力的收敛性，同时前者用来帮IPM转向。内力环的出现改进了控制性能，并且这对于悬浮来说是基本的。给定不稳定的力平衡，这是重要的在尝试IPM转向前保证内环的稳定性。
We take into account a back-stepping approach on two levels (or loops): pose loop and force loop. The latter, considered as an internal loop, is designed to guarantee the convergence of the actual force on the IPM to the desired one, while the former aims to steer the IPM. The presence of the internal force loop improves the control properties, compared to previous approaches, and it is fundamental for levitation. Given the unstable force equilibrium, it is essential to guarantee the stability of this internal loop before attempting to steer the IPM.

在分析下的具体情形下，尾绳是有益的因为它作为一个沿重力方向的动力学的稳定阻尼，提高了系统的稳定性。没有任何限制将所提出的理论应用于无绳胶囊中，但是我们期望需要一个更快的控制环来处理低阻尼动力学。
In the specific case under analysis, the tether is beneficial as it acts as a stabilizing damper on the dynamics along the gravity direction, improving stability in the system. There is no limitation in applying the proposed method to untethered capsules, but we expect the need for a faster control loop to handle the less damped dynamics.

胶囊的名义动力学 The nominal dynamics of the capsule:

$$B(x)\ddot{x}+C(x,\dot{x})\dot{x}+G(x)={\tau }_m(x,q)$$

以上方程的导数 the time derivation of this function：

$$\dot{{\tau }_m}=\frac{\partial \tau (x,q)}{\partial x}\dot{x}+\frac{\partial \tau (x,q)}{\partial q}\dot{q}=J_x\dot{x}+J_q\dot{q}$$

将${\tau }_m$作为我们要去控制的系统的状态变量，$\dot{q}$作为控制输入 turn ${\tau }_m$ into a state variable for the system we aim to control and $\dot{q}$ into the control input.

整体动力学：

$$\{\begin{array}{r}B(x)\ddot{x}+C(x,\dot{x})\dot{x}+G(x)=\tau \\ \dot{\tau }=J_x\dot{x}+J_q\dot{q}+\dot{v}\end{array}$$

3.A pose control

定义一个位姿控制器，尝试把IPM转向到期望轨迹，这控制器是第一步，并且通过首先考虑$\tau$能被故意设定为一个上面动力学的控制输入获得。
Defining a pose controller that attempts to steer the IPM to a desired trajectory is the first step and is achieved by first considering that $\tau$ can be deliberately set as a control input for the upper dynamics.

$${\tau }_d=G(x)+K_p\tilde{x}+K_d\dot{\tilde{x}}$$
$$\tilde{x}=x_d-x$$

假设1：下面介绍的力/力矩环的执行更快，几乎是瞬间的。期望的轨迹是时间的分段恒定函数。
在第一个假设下，对于任何正定的$K_p$和$K_d$，位姿控制器获得误差$\tilde{x}$的渐近稳定性。

3.B force control

为了设计一个用于力和力矩的渐进稳定性控制器，我们考虑(2)并搜索$\dot{q}$，这样用于$\tilde{\tau }={\tau }_d-{\tau }_m$的动力学发展为$\dot{\tilde{\tau }}=-K\tilde{\tau }$，使用K正定设计增益。这导致力和力矩误差动力学的渐进稳定性。
In order to design an asymptotically stable controller for force and torque, we take into account (2) and search for $\dot{q}$ so that the dynamics for $\tilde{\tau }={\tau }_d-{\tau }_m$ evolves as $\dot{\tilde{\tau }}=-K\tilde{\tau }$, with $K$ positive definite design gain. This leads to asympototic stability of the force and torque error dynamics.

$$\dot{{\tau }_d}-\dot{{\tau }_m}=-K\tilde{\tau }$$

$$\dot{{\tau }_d}-J_x\dot{x}-J_q\dot{q}=-K\tilde{\tau }$$

$$\dot{q}=J_q^{†}(\dot{{\tau }_d}+K\tilde{\tau }-J_x\dot{x})$$

在假定干扰$v$接近0的情况下，任何正定$K$都可以使力矩动力学稳定。

3.C Overall control

$$\{\begin{array}{r}{\tau }_d=G(x)+K_p\tilde{x}+K_d\dot{\tilde{x}}\\ \dot{q}=J_q^{†}(\dot{{\tau }_d}+K\tilde{\tau }-J_x\dot{x}-\dot{x})\end{array}$$

弱化假设1，$\dot{\tilde{\tau }}=-K\tilde{\tau }+\dot{x}$

假设2：$x_d$是时间的分段恒定函数，且$v$接近0.

在假设2下，上面的控制器对于任何正定增益$K_p$ $K_d$ $K$都有渐近稳定性。

[1]: Pittiglio, Giovanni, et al. “Magnetic levitation for soft-tethered capsule colonoscopy actuated with a single permanent magnet: a dynamic control approach.” IEEE robotics and automation letters 4.2 (2019): 1224-1231.