## ME327 Pantagraphic Analysis课程简介

Allison Okamura is indeed a Professor in Mechanical Engineering at Stanford University, with a research focus on haptics and medical robotics. Brandon Ritter and Zonghe Chua were both listed as course assistants for ME 327 during the Spring 2020 semester, and are indeed graduate students in Mechanical Engineering at Stanford University.

## PREREQUISITES

Welcome to ME 327: Design and Control of Haptic Systems. The CURRENT offering of this course is in Spring 2022 and course materials for this year should be accessed by Stanford students through Canvas. In contrast, this webpage is offered as a service for the worldwide haptics research and teaching community and includes only course materials from when the class was offered online in Spring 2020. The next offering of ME327 at Stanford will be in Spring 2023. You can also access a lower-level, self-paced MOOC for haptics here: https://hapticsonline.class.stanford.edu/ (free to watch videos, pay for quizzes, assignments, and a certificate). Note that due to changes in availability of parts, the current Hapkit (lab kit) parts list and instructions are out of date. We aim to update these in Summer 2022.

In this class, we study the design and control of haptic systems, which provide touch feedback to human users interacting with virtual environments and teleoperated robots. This class is aimed toward graduate students and advanced undergraduates in engineering and computer science. This class requires a background in dynamic systems and programming. Experience with feedback control and mechanical prototyping is also useful. Course information and policies for 2020 are contained in the syllabus ब. This course covers device modeling (kinematics and dynamics), synthesis and analysis of control systems, design and implementation of mechatronic devices, and human-machine interaction.

## ME327 Pantagraphic Analysis HELP（EXAM HELP， ONLINE TUTOR）

Show that equality $|(x \mid y)|=|x||y|$ for the Cauchy-Schwarz inequality holds if and only if $x$ and $y$ are collinear (that is, both belong to some one-dimensional subspace).

Find the transfer function $G_2(s)=Y_h(s) / F_{u p}(s)$, where $y_h(t)$ is the pantograph head displacement.

Give an example of a nonempty compact convex set $C$ in a two-dimensional Banach space $X$ along with a vector $x \in X$ such that the set
$$\left{c \in C:|x-c|=\min _{y \in C}|x-y|\right}$$
consists of more than one element.

In this problem we determine some orthogonal complements.
(a) Let
$$Y:=\left{f \in L^2(0,1): f(t)=0 \text { for almost all } t \in\left(0, \frac{1}{2}\right)\right}$$
Show that $Y$ is a closed subspace of $L^2(0,1)$ and find $Y^{\perp}$.
(b) Let
$$Y:=\left{f \in L^2(0,1): \int_0^1 f(t) \mathrm{d} t=0\right} .$$
Show that $Y$ is a closed subspace of $L^2(0,1)$ and find $Y^{\perp}$.

## Textbooks

• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.

Statistics-lab™可以为您提供stanford.eduME327 Pantagraphic Analysis泛函分析的代写代考和辅导服务！ 请认准Statistics-lab™. Statistics-lab™为您的留学生涯保驾护航。

## ME327 Pantagraphic Analysis课程简介

Allison Okamura is indeed a Professor in Mechanical Engineering at Stanford University, with a research focus on haptics and medical robotics. Brandon Ritter and Zonghe Chua were both listed as course assistants for ME 327 during the Spring 2020 semester, and are indeed graduate students in Mechanical Engineering at Stanford University.

## PREREQUISITES

Welcome to ME 327: Design and Control of Haptic Systems. The CURRENT offering of this course is in Spring 2022 and course materials for this year should be accessed by Stanford students through Canvas. In contrast, this webpage is offered as a service for the worldwide haptics research and teaching community and includes only course materials from when the class was offered online in Spring 2020. The next offering of ME327 at Stanford will be in Spring 2023. You can also access a lower-level, self-paced MOOC for haptics here: https://hapticsonline.class.stanford.edu/ (free to watch videos, pay for quizzes, assignments, and a certificate). Note that due to changes in availability of parts, the current Hapkit (lab kit) parts list and instructions are out of date. We aim to update these in Summer 2022.

In this class, we study the design and control of haptic systems, which provide touch feedback to human users interacting with virtual environments and teleoperated robots. This class is aimed toward graduate students and advanced undergraduates in engineering and computer science. This class requires a background in dynamic systems and programming. Experience with feedback control and mechanical prototyping is also useful. Course information and policies for 2020 are contained in the syllabus ब. This course covers device modeling (kinematics and dynamics), synthesis and analysis of control systems, design and implementation of mechatronic devices, and human-machine interaction.

## ME327 Pantagraphic Analysis HELP（EXAM HELP， ONLINE TUTOR）

Find the transfer function, $G_1(s)=Y_{c a t}(s) / F_{u p}(s)$, where $y_{c a t}$ is the catenary displacement and $f_{u p}(t)$ is the upward force applied to the pantograph under active control.

To find the transfer function, we start with the equation of motion for the catenary:

\frac{\partial^2 y}{\partial x^2} = \frac{T}{\rho} \sqrt{1 + \left(\frac{\partial y}{\partial x}\right)^2}∂x2∂2y​=ρT​1+(∂x∂y​)2​

where $y(x)$ is the displacement of the catenary in the $y$ direction, $T$ is the tension in the catenary, $\rho$ is the weight per unit length of the catenary, and $x$ is the distance along the catenary.

Let’s assume that the upward force $F_{up}$ is applied at a single point on the catenary, and that the catenary is otherwise fixed at both ends. We can write the equation of motion for this system as:

\frac{\partial^2 y}{\partial x^2} = \frac{T}{\rho} \sqrt{1 + \left(\frac{\partial y}{\partial x}\right)^2} + \frac{F_{up}}{\rho} \delta(x – x_{up})∂x2∂2y​=ρT​1+(∂x∂y​)2​+ρFup​​δ(x−xup​)

where $\delta(x-x_{up})$ is the Dirac delta function that models the force applied at the point $x_{up}$.

Taking the Laplace transform of both sides of this equation, assuming zero initial conditions, we get:

s^2 Y_{cat}(s) = \frac{T}{\rho} \int_{-\infty}^\infty \sqrt{1 + \left(\frac{\partial y}{\partial x}\right)^2} e^{-s x/\sqrt{1+(\partial y/\partial x)^2}} dx + \frac{F_{up}}{\rho} e^{-s x_{up}}s2Ycat​(s)=ρT​∫−∞∞​1+(∂x∂y​)2​e−sx/1+(∂y/∂x)2​dx+ρFup​​e−sxup​

We can simplify the integral using the chain rule:

\frac{\partial}{\partial x} \left(\frac{\partial y}{\partial x}\right) = \frac{T}{\rho} \frac{\partial}{\partial x} \left(\sqrt{1 + \left(\frac{\partial y}{\partial x}\right)^2}\right)∂x∂​(∂x∂y​)=ρT​∂x∂​⎝⎛​1+(∂x∂y​)2​⎠⎞​

\Rightarrow \sqrt{1 + \left(\frac{\partial y}{\partial x}\right)^2} d\left(\frac{\partial y}{\partial x}\right) = \frac{T}{\rho} dx⇒1+(∂x∂y​)2​d(∂x∂y​)=ρT​dx

Integrating both sides, we get:

\int_{\frac{\partial y}{\partial x}(0)}^{\frac{\partial y}{\partial x}(x)} \sqrt{1 + \left(\frac{\partial y}{\partial x}\right)^2} d\left(\frac{\partial y}{\partial x}\right) = \frac{T}{\rho} x∫∂x∂y​(0)∂x∂y​(x)​1+(∂x∂y​)2​d(∂x∂y​)=ρT​x

This simplifies to:

\frac{\partial y}{\partial x} = \sqrt{\left(\frac{T}{\rho}\right)^2 x^2 + C}∂xy​=(ρT​)2x2+C

where $C$ is a constant of integration. To determine the value of $C$, we use the boundary condition that the catenary is fixed at both ends. Let the length of the catenary be $L$ and let the height at the ends be $h$. Then we have:

\int_0^L \sqrt{1 + \left(\frac{\partial y}{\partial x}\right)^2} dx = \int_0^h \frac{dy}{\sqrt{h^2 – y^2}}∫0L​1+(∂x∂y​)2​dx=∫0h​h2−y2​dy​

which is the integral for the length of a circle of radius $h$.

Find the transfer function $G_2(s)=Y_h(s) / F_{u p}(s)$, where $y_h(t)$ is the pantograph head displacement.

To find the transfer function $G_2(s) = Y_h(s) / F_{up}(s)$, we can start with the equation of motion for the pantograph head:

M_h \frac{d^2 y_h}{dt^2} + C_h \frac{dy_h}{dt} + K_h y_h = F_{up}Mh​dt2d2yh​​+Ch​dtdyh​​+Kh​yh​=Fup​

where $M_h$, $C_h$, and $K_h$ are the mass, damping coefficient, and stiffness of the pantograph head, respectively.

Taking the Laplace transform of both sides of this equation, assuming zero initial conditions, we get:

M_h s^2 Y_h(s) + C_h s Y_h(s) + K_h Y_h(s) = F_{up}(s)Mh​s2Yh​(s)+Ch​sYh​(s)+Kh​Yh​(s)=Fup​(s)

Dividing both sides by $F_{up}(s)$, we get:

\frac{Y_h(s)}{F_{up}(s)} = \frac{1}{M_h s^2 + C_h s + K_h} = \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}Fup​(s)Yh​(s)​=Mh​s2+Ch​s+Kh​1​=s2+2ζωn​s+ωn2​ωn2​​

where $\omega_n = \sqrt{K_h/M_h}$ is the natural frequency of the pantograph head and $\zeta = C_h/(2\sqrt{M_h K_h})$ is the damping ratio.

Therefore, the transfer function $G_2(s)$ can be expressed in the form of a second-order system with natural frequency $\omega_n$ and damping ratio $\zeta$.

## Textbooks

• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.