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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 Note that due to changes in availability of parts, the current Hapkit (lab kit) parts list and instructions are out of date.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）

Proof. We prove that $\operatorname{int}(K)$ is convex. Let $x_0, x_1 \in \operatorname{int}(K)$, choose a real number $0<\lambda<1$, and define $x_\lambda:=(1-\lambda) x_0+\lambda x_1$. Choose open sets $U_0, U_1 \subset X$ such that $x_0 \subset U_0 \subset K$ and $x_1 \subset U_1 \subset K$ and define
$$
U:=\left(U_0-x_0\right) \cap\left(U_1-x_1\right)=\left{x \in X \mid x_0+x \in U_0, x_1+x \in U_1\right} .
$$
Then $U \subset X$ is an open set containing the origin such that $x_0+U \subset K$ and $x_1+U \subset K$. Since $K$ is convex, this implies that $x_\lambda+U$ is an open subset of $K$ containing $x_\lambda$. Hence $x_\lambda \in \operatorname{int}(K)$.

We prove that $\bar{K}$ is convex. Let $x_0, x_1 \in \bar{K}$, choose a real number $0<\lambda<1$, and define $x_\lambda:=(1-\lambda) x_0+\lambda x_1$. Let $U$ be an open neighborhood of $x_\lambda$. Then the set
$$
W:=\left{\left(y_0, y_1\right) \in X \times X \mid(1-\lambda) y_0+\lambda y_1 \in U\right}
$$
is an open neighborhood of the pair $\left(x_0, x_1\right)$, by continuity of addition and scalar multiplication. Hence there exist open sets $U_0, U_1 \subset X$ such that
$$
x_0 \in U_0, \quad x_1 \in U_1, \quad U_0 \times U_1 \subset W .
$$
Since $x_0, x_1 \in \bar{K}$, the sets $U_0 \cap K$ and $U_1 \cap K$ are nonempty. Choose elements $y_0 \in U_0 \cap K$ and $y_1 \in U_1 \cap K$. Then $\left(y_0, y_1\right) \in U_0 \times U_1 \subset W$ and hence $y_\lambda:=(1-\lambda) y_0+\lambda y_1 \in U \cap K$. Thus $U \cap K \neq \emptyset$ for every open neighborhood $U$ of $x_\lambda$ and so $x_\lambda \in \bar{K}$.

We prove the last assertion. Assume $\operatorname{int}(K) \neq \emptyset$ and fix an element $x \in K$. Then the set $U_x:={t x+(1-t) y \mid y \in \operatorname{int}(K), 0<t<1}$ is open and contained in $K$. Hence $U_x \subset \operatorname{int}(K)$ and so $x \in \bar{U}_x \subset \overline{\operatorname{int}(K)}$. This proves Lemma 3.10.

Proof. Assume first that $B={0}$. Then the set
$$
P:={t x \mid x \in A, t \geq 0}
$$
satisfies the conditions (P1), (P2), (P3) on page 79. Hence $(X, \preccurlyeq)$ is an ordered vector space with the partial order defined by $x \preccurlyeq y$ iff $y-x \in P$. Let $x_0 \in A$. Then the linear subspace $Y:=\mathbb{R} x_0$ satisfies (O3) on page 76 . Hence Theorem 2.38 asserts that there exists a positive linear functional $\Lambda: X \rightarrow \mathbb{R}$ such that $\Lambda\left(x_0\right)=1$. If $x \in A$ then $x-t x_0 \in A$ for $t>0$ sufficiently small because $A$ is open and hence $\Lambda(x) \geq t>0$.
We prove that $\Lambda$ is continuous. To see this, define
$$
U:={x \in X \mid \Lambda(x)>0}
$$
and fix an element $x \in U$. Then
$$
V:=\left{y \in X \mid x_0+\Lambda(x)^{-1}(y-x) \in A\right}
$$
is an open set such that $x \in V \subset U$. This shows that $U$ is an open set and hence, so is the set
$$
\Lambda^{-1}((a, b))=\left(a x_0+U\right) \cap\left(b x_0-U\right)
$$
for every pair of real numbers $a<b$. Hence $\Lambda$ is continuous and this proves the result for $B={0}$.

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.

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