Lecture1 Introduction

Introduction of robotics

1. reference textbook:
   1. Cai ZiXing,2015
   1. Saeed B. Niku.,2013
   2. Siciliano,B.,2010

Types of robots

$$robot\{\begin{array}{l}Robotmanipulators\\ Mobilerobots\{\begin{array}{l}Groundrobots\{\begin{array}{l}Wheeledrobots\\ Leggedrobots\end{array}\\ Submarinerobots\\ Aerialrobots\end{array}\end{array}$$

Lecture2 Manufacturing Systems & Robot Kinematics

1. manufacturing system

1.1 manufacturing classification

1. project
   1. one of a kind(ship & nuclear power plant) and complex and site building
2. job shop
   1. not site building and low volume and production quantities
3. repetitive
   1. fixed routing,repeat
4. line
   1. delivery time required is short,require many different models and options
5. continuous
   1. volume is high…,like chemical components

2. kinematicas of robot manipulator

2.1 robot manipulators

2.1.1 configuration

1. robot arms
   1. rigid bodies(links) connected by joints
   2. joints:
      1. revolute, prismatic, spherical
   3. drive:
      1. electric or hydraulic
   4. end-effector(tool):
      1. mounted on a flange or plate secured to the wrist joint of robot
2. robot configuration
   1. cartesian(笛卡尔): PPP
   2. cylindrical(圆柱): RPP
   3. articulated(铰接): RRP
   4. spherical(球坐标): RRR
   5. SCARA: RRP(selective compliance assembly robot arm)
   6. hand coordinate
3. motion control methods
   1. point to point control
      1. a sequence of discrete points
      2. like: spot welding(点焊), pick-and-place, loading and unloading
   2. continuous path control
      1. follow a prescribed path, controlled-path motion
      2. like: spray painting, Arc welding(弧焊), gluing

2.1.2 robot specifications

specification

1. number of axes:
   1. major axes,(1-3)=>position the wrist
   2. minor axes,(4-6)=>orient the rool
   3. redundant,(7-n)=>reaching around obstacles, avoiding undesirable configuration
2. degree of freedom(DOF)
3. workspace
4. payload(load capacity)
5. precision v.s. repeatability(精度和集中度)

DOF

1. The Chebychev-Grübler-Kutzbach formula:
   1. $$M=d(n-g-1)+\sum _{i=1}^g{f}_i$$
   • M: the mobility or the system DOF
   • d: the order of the sysem (d=3 for planar motion,and d=6 for spatial motion)
   • n: the number of the links including the frames
   • g: the number of joints
   • $f_i$: the number of DOFs for the ith joint
2. DOF of joints
   1. cylindrical joint (1R,1P):DOF=2 , 2 components
   2. prismatic joint (1P):DOF=1 , 2 components
   3. ball(or spherical) joint: DOF=3 , 1 components
   4. Hooker joint(or U(universe)-jont) : DOF=2 ,1 components
   5. revolute joint: DOF=1 ,2 components

2.2 kinematics

2.2.1 what is kinematics

1. forward kinematics:
   1. given joint variables, find end-effection position
2. inverse kinematics
   1. given end-effectation position and orientation, find joint variables

2.2.2 preliminary

robot reference frames

1. world frame -> fixed
2. joint frame
3. tool frame

coordinate transformation

rotation only

1. reference coordinate f
2. rame OXYZ
3. body-attached frame O’uvw
   point represented in OXYZ is $P_{xyz}=[p_x,p_y,p_z{]}^T$, or ${\mathbf{p}}_{xyz}=p_x{\mathbf{i}}_x+p_y{\mathbf{j}}_y+p_z{\mathbf{k}}_z$
   point represented in O’uvw:
   ${\mathbf{p}}_{uvw}=p_u{\mathbf{i}}_u+p_v{\mathbf{j}}_v+p_w{\mathbf{k}}_w$

---

1. uvw to XYZ coordinate transformation
   • rotation only: ${\mathbf{p}}_{xyz}=R{\mathbf{p}}_{uvw}$
     we new: $p_x=\mathbf{i}\cdot \mathbf{p}$ , so ${\mathbf{p}}_{xyz}=\left[\begin{array}{c}{\mathbf{i}}_x^T\\ {\mathbf{j}}_y^T\\ {\mathbf{k}}_z^T\end{array}\right]\cdot \mathbf{p}$
     where, $\mathbf{p}=\left[\begin{array}{ccc}{\mathbf{i}}_u& {\mathbf{j}}_v& {\mathbf{k}}_w\end{array}\right]\left[\begin{array}{c}{p}_u\\ p_v\\ p_w\end{array}\right]$
     so, that$$\left[\begin{array}{c}{p}_x\\ p_y\\ p_z\end{array}\right]=\left[\begin{array}{ccc}{\mathbf{i}}_x\cdot {\mathbf{i}}_u& {\mathbf{i}}_x\cdot {\mathbf{j}}_v& {\mathbf{i}}_x\cdot {\mathbf{k}}_w\\ {\mathbf{j}}_y\cdot {\mathbf{i}}_u& {\mathbf{j}}_y\cdot {\mathbf{j}}_v& {\mathbf{j}}_y\cdot {\mathbf{k}}_w\\ {\mathbf{k}}_z\cdot {\mathbf{i}}_u& {\mathbf{k}}_z\cdot {\mathbf{j}}_v& {\mathbf{k}}_z\cdot {\mathbf{k}}_w\end{array}\right]\cdot \left[\begin{array}{c}{p}_u\\ p_v\\ p_w\end{array}\right]$$
     so so that:
     • Rot(x,$\theta$)=$\left[\begin{array}{ccc}1& 0& 0\\ 0& \cos \theta & -\sin \theta \\ 0& \sin \theta & \cos \theta \end{array}\right]$
     • Rot(y,$\theta$)=$\left[\begin{array}{ccc}\cos \theta & 0& \sin \theta \\ 0& 1& 0\\ -\sin \theta & 0& \cos \theta \end{array}\right]$
     • Rot(z,$\theta$)=$\left[\begin{array}{ccc}\cos \theta & -\sin \theta & 0\\ \sin \theta & \cos \theta & 0\\ 0& 0& 1\end{array}\right]$
       Matrix R 可看作转换矩阵，也可看做对向量做旋转操作的旋转矩阵
2. XYZ to uvw coordinate transformation
   to find matrix Q:let ${\mathbf{p}}_{uvw}=Q\cdot {\mathbf{p}}_{xyz}$
   similarly that:
   $$\left[\begin{array}{c}{p}_u\\ p_v\\ p_w\end{array}\right]=\left[\begin{array}{ccc}{\mathbf{i}}_u\cdot {\mathbf{i}}_x& {\mathbf{i}}_u\cdot {\mathbf{j}}_y& {\mathbf{i}}_u\cdot {\mathbf{k}}_z\\ {\mathbf{j}}_v\cdot {\mathbf{i}}_x& {\mathbf{j}}_v\cdot {\mathbf{j}}_y& {\mathbf{j}}_v\cdot {\mathbf{k}}_z\\ {\mathbf{k}}_w\cdot {\mathbf{i}}_x& {\mathbf{k}}_w\cdot {\mathbf{j}}_y& {\mathbf{k}}_w\cdot {\mathbf{k}}_z\end{array}\right]\cdot \left[\begin{array}{c}{p}_x\\ p_y\\ p_z\end{array}\right]$$
   we can figure out that:
   $Q=R^{-1}=R^T$
   Matrix Q 可看作转换矩阵，也可看做对向量做与R相反$\theta$的旋转操作的旋转矩阵

---

composite rotation matrix

1. a sequence of finite rotations:
   1. matrix multiplications do not commute
   2. rules:
      • if rotating coordinate O-U-V-W is rotating about principal axis of OXYZ frame,then pre-multiply the R to the I
      • if rotating coordinate O-U-V-W is rotating about principal axis of OUVW frame,then post-multiply the R to the I

Homogeneous representation

1. coordiante transformation from {B} to {A}:
   ${}^A{r}^P{=}^A{R}_B{\cdot }^B{r}^P{+}^A{r}^{O^{\prime }}$
2. Homogeneous transformation matrix:
   ${}^A{T}_B=\left[\begin{array}{cc}{}^A{R}_B& {}^A{r}^{O^{\prime }}\\ 0_{1\times 3}& 1\end{array}\right]$
   $\left[\begin{array}{c}{}^A{r}_P\\ 1\end{array}\right]{=}^A{T}_B\cdot \left[\begin{array}{c}{}^B{r}_P\\ 1\end{array}\right]$

---

composite Homogeneous transformation matrix

• similar to composite rotation matrix

Orientation Representation

1. Euler angles reoresentation ($\varphi$, $\theta$, $\psi$ )
   • many different typses
     • Euler angle I:
       • $\varphi$: about OZ axis
       • $\theta$: about OU axis
       • $\psi$: about OW axis
     • Euler angle II
       • $\varphi$: about OZ axis
       • $\theta$: about OV axis
       • $\psi$: about OW axis
     • Roll-Pitch-Yaw
       • $\varphi$: about OX axis
       • $\theta$: about OY axis
       • $\psi$: about OZ axis
   • description of Euler angle representations

Lecture3 Mathematical Background

3.1 Matrix

• symmetric matrix:
  $A^T=A\Rightarrow a_{kj}=a_{jk}$
• skew-symmetric matrix:
  $A^T=-A\Rightarrow a_{kj}=-a_{jk}$
• orthogonal matrix:
  $A^T=A^{-1}$
  any real square matrix A may be written as the sum of a symmetric matrix R and a skew-symmetric matrix S,where:$R=\frac{1}{2}(A+A^T)S=\frac{1}{2}(A-A^T)$
• triangular matrix(上\下三角)
• diagonal matrix（对角矩阵）
• scalar matrix $\left[\begin{array}{ccc}a& 0& 0\\ 0& a& 0\\ 0& 0& a\end{array}\right]$
• unit matrix(单位)
• inner product or dot product:
  $\mathbf{a}\cdot \mathbf{b}={\mathbf{a}}^T\mathbf{b}$
• idempotent matrix:
  $A^2=A$
• orthogonality:
  $\mathbf{a}\cdot \mathbf{b}=0\Rightarrow \mathbf{a}\text{ and }\mathbf{b}\text{ is orthogonality}$
• vector product:
  $V=\mathbf{a}\times \mathbf{b}=[a_2{b}_3-a_3{b}_2,a_3{b}_1-a_1{b}_3,a_1{b}_2-a_2{b}_1]$
  $=\left|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ a_1& a_2& a_3\\ b_1& b_2& b_3\end{array}\right|$
  $|V|=|a||b|\sin \gamma$
• scalar triple product or mixed triple product:
  $\mathbf{a}=[a_1,a_2,a_3],\mathbf{b}=[b_1,b_2,b_3]\mathbf{c}=[c_1,c_2,c_3]$
  $(\mathbf{a}\mathbf{b}\mathbf{c})=\mathbf{a}\cdot (\mathbf{b}\times \mathbf{c})=\left|\begin{array}{ccc}{a}_1& a_2& a_3\\ b_1& b_2& b_3\\ c_1& c_2& c_3\end{array}\right|$
• tr(A):
  $tr(A)={\sum }_{i=1}^3{a}_{ii}$
• Vert(A)
  $Vert(A)=\frac{1}{2}\left[\begin{array}{c}{a}_{32}-a_{23}\\ a_{13}-a_{31}\\ a_{21}-a_{12}\end{array}\right]$
• linear transformations:
  • isomorphism: a linear transformation of a vector space $\gamma$ into itself
  • linear transformation of the 3D space
    • projections
    • reflections
    • rotation
    • affine(仿射)
• properties of orthogonal projection P of ${\epsilon }^3$($R^3$):
  $$P^2=P\text{ (idempotent)}$$ $$P\cdot \mathbf{n}=0\text{ n: unit normal}$$
• orthogonal projection: P onto $\Pi$
  $$P=1-\mathbf{n}{\mathbf{n}}^T$$
• reflection R of ${\epsilon }^3$ onto a plane $\Pi$passing through the origin:
  $$R=1-2\mathbf{n}{\mathbf{n}}^T$$
• component $\mathbf{v}$ along $\mathbf{e}$ and normal ${\mathbf{v}}_e$
  $${\mathbf{v}}_e=\mathbf{e}{\mathbf{e}}^T\mathbf{v}$$
  $${\mathbf{v}}_n=(1-\mathbf{e}{\mathbf{e}}^T)\mathbf{v}$$
  $\mathbf{e}$: unit vector
• L’s eigenvector $\lambda \mathbf{e}$ and eigenvalue $\lambda$
  $$det(\lambda I-L)=0$$
  (characteristic equation)

3.2 rigid-body rotations

• distance between $\mathbf{u}$ and $\mathbf{v}$:
  $d=\sqrt{(\mathbf{u}-\mathbf{v}{)}^T\cdot (\mathbf{u}-\mathbf{v})}$
• the volume V of the tetrahedron(四面体) defined by the origin and these points $\mathbf{u},\mathbf{v},\mathbf{w}$:
  $V=\frac{1}{6}|\mathbf{u}\times \mathbf{v}\cdot \mathbf{w}|=\frac{1}{6}|det[\mathbf{u}\mathbf{v}\mathbf{w}]|$
• Let Q be an isometry mapping the triad {$\mathbf{u},\mathbf{v},\mathbf{w}$} into {${\mathbf{u}}^{\prime },{\mathbf{v}}^{\prime },{\mathbf{w}}^{\prime }$}, the distance from the origin to the points vector $\mathbf{u},\mathbf{v},\mathbf{w}$ is:
  $||\mathbf{u}||=\sqrt{\mathbf{u}{\mathbf{u}}^T},||\mathbf{v}||=\sqrt{\mathbf{v}{\mathbf{v}}^T},||\mathbf{w}||=\sqrt{\mathbf{w}{\mathbf{w}}^T}$
  Clearly:
  $||\mathbf{u}||=||{\mathbf{u}}^{\prime }||,||\mathbf{v}||=||{\mathbf{v}}^{\prime }||,||\mathbf{w}||=||{\mathbf{w}}^{\prime }||$
  $det[{\mathbf{u}}^{\prime },{\mathbf{v}}^{\prime },{\mathbf{w}}^{\prime }]=\pm det[\mathbf{u},\mathbf{v},\mathbf{w}]$
  case1: sign is preserved, the isometry is rotation
  case2: otherwise, isometry represents a reflection.
  Hence,distance preservation requires
  ${\mathbf{p}}^T\mathbf{p}={\mathbf{p}}^{\prime T}{\mathbf{p}}^{\prime }$, where ${\mathbf{p}}^{\prime }=Q\mathbf{p}$
  $\Rightarrow Q^{T}Q=I\Rightarrow Q:\text{orthogonal matrix}$
  let $T=[\mathbf{u}\mathbf{v}\mathbf{w}],T^{\prime }=[{\mathbf{u}}^{\prime }{\mathbf{v}}^{\prime }{\mathbf{w}}^{\prime }]$
  we have $T^{\prime }=QT$, for a rigid-body matrix:
  $det(T)=det(T^{\prime })\Rightarrow det(Q)=\pm 1,\text{ proper othogonal matrix}$
• the cross-product matrix(矩阵叉乘):
  $V=\left[\begin{array}{c}{v}_1\\ v_2\\ v_3\end{array}\right]$, $\mathbf{x}=\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]$
  $V\times \mathbf{x}=\left[\begin{array}{c}{v}_2{x}_3-v_3{x}_2\\ v_3{x}_1-v_1{x}_3\\ v_1{x}_2-v_2{x}_1\end{array}\right]$
  $\frac{\partial (V\times \mathbf{x})}{\partial \mathbf{x}}=[\frac{\partial (V\times \mathbf{x})}{\partial {\mathbf{x}}_1}\frac{\partial (V\times \mathbf{x})}{\partial {\mathbf{x}}_2}\frac{\partial (V\times \mathbf{x})}{\partial {\mathbf{x}}_3}]=\left[\begin{array}{ccc}0& -v_3& v_2\\ v_3& 0& -v_1\\ -v_2& v_1& 0\end{array}\right]$
  given any 3-D vector $\mathbf{a}$, then its cross-product matrix $\mathbf{A}$ is skew-symmetric matrix $A^T=-A$
• rotation matrix $\mathbf{Q}$:

  1. $Q=\mathbf{e}{\mathbf{e}}^T+\cos \varphi (1-\mathbf{e}{\mathbf{e}}^T)+\sin \varphi E$
     $[E]=1\times \mathbf{e}=\left[\begin{array}{ccc}0& -e_3& e_2\\ e_3& 0& -e_1\\ -e_2& e_1& 0\end{array}\right]$

  2. an alternative representation Q could be:
     $Q=1+\sin \varphi E+(1-\cos \varphi )E^2$

  3. the canonical form of the rotation matrix:
     $[Q{]}_x=\left[\begin{array}{ccc}1& 0& 0\\ 0& \cos \varphi & -\sin \varphi \\ 0& \sin \varphi & \cos \varphi \end{array}\right]$, $[Q{]}_y=\left[\begin{array}{ccc}\cos \varphi & 0& \sin \varphi \\ 0& 1& 0\\ -\sin \varphi & 0& \cos \varphi \end{array}\right]$

     $[Q{]}_z=\left[\begin{array}{ccc}\cos \varphi & -\sin \varphi & 0\\ \sin \varphi & \cos \varphi & 0\\ 0& 0& 1\end{array}\right]$

• the linear invariant of a rotation(已知旋转矩阵求$\mathbf{e}$和$\varphi$):
  1. $\mathbf{q}=Vert(Q)=Vert(\sin \varphi \mathbf{E})=\sin \varphi \mathbf{e}$
  2. $tr(Q)=1+2\cos \varphi$
  3. $\sin \varphi =\pm ||\mathbf{q}||$
  4. $\mathbf{e}=\frac{\mathbf{q}}{\sin \varphi }$
  5. $\cos \varphi =\frac{tr(Q)-1}{2}=q_0$
  6. $Q=\frac{\mathbf{q}{\mathbf{q}}^T}{||\mathbf{q}|{|}^2}+q_0(1-\frac{\mathbf{q}{\mathbf{q}}^T}{||\mathbf{q}|{|}^2})+\overline{Q}$
• properties of 3 linear transformation matrix :
  1. orthogonal:
     1. idempotent matrix
     2. singular: $det(Q)=0$
     3. symmetric matrix
  2. rotation:
     1. $Q{Q}^T=1$
     2. $det(Q)=1$
  3. reflection:
     1. symmetric matrix
     2. $tr(Q)\ne -1,3$
     3. $Q{Q}^T=1$
     4. $det(Q)=-1$

Lecture4 Fundamentals of Rigid-Body Mechanics

angular velcity matrix of rigid-body motion

$\mathbf{p}(t)=Q(t)\cdot {\mathbf{p}}_o\Rightarrow \dot{\mathbf{p}}(t)=\dot{Q}(t){\mathbf{p}}_{o}and{\mathbf{p}}_o=Q^T(t)\mathbf{p}(t)$
so that, $\dot{\mathbf{p}}(t)=\dot{Q}{Q}^T\mathbf{p}$
let :
$\Omega =\dot{Q}{Q}^T$
where $\omega =\left[\begin{array}{c}{\omega }_x\\ {\omega }_y\\ {\omega }_z\end{array}\right]$ and $\Omega =1\times \omega$

general instantaneous motion of a rigid-body

• position :
  $\mathbf{p}(t)=\mathbf{a}(t)+Q(t)(\mathbf{p}-\mathbf{a})$
• velcity :
  $\dot{\mathbf{p}}=\dot{\mathbf{a}}+\Omega (\mathbf{p}-\mathbf{a})$
  Subsequently, $(\dot{\mathbf{p}}-\dot{\mathbf{a}})(\mathbf{p}-\mathbf{a})=0$

it indicates the relative velocity of two point of the same rigid body is perpendicular(垂直) to the line joining them.

hence, the instantaneous motion of a body is equivalent to that of bolt of a screw of axis L’,called Instantaneous Screw Axis(ISA)(旋转轴)

As the ISA changes, the motion of the body is called an instantaneous screw
${\mathbf{v}}_0=v_0\frac{\omega }{||\omega ||}$, $\omega$ called amplitude
p’: pitch of the instantaneous screw
$p^{\prime }=\frac{v_0}{||\omega ||}m/rad$

to determine the set of the points of the plate that undergo a velocity of minimum magnitude:
$p_o^{{}^{\prime }}=\frac{1}{||\omega ||}\Omega \dot{C}$

* ISA L’ can be specified uniquely through its Pl$\ddot{u}$cker(普吕克) array ${\mathbf{p}}_L$:

${\mathbf{p}}_L^{\prime }=\left[\begin{array}{c}{\mathbf{e}}^{\prime }\\ {\mathbf{n}}^{\prime }\end{array}\right]$
where ${\mathbf{e}}^{\prime }$: direction of L’ and ${\mathbf{n}}^{\prime }$ : moment about origin
and ${\mathbf{n}}^{\prime }={\mathbf{p}}^{\prime }\times {\mathbf{e}}^{\prime }$
${\mathbf{e}}^{\prime }=\frac{\omega }{||\omega ||}$

Twist-transfer Formula

it relates the twist of the same rigid body at two different point
${\mathbf{t}}_A=\left[\begin{array}{c}\omega \\ {\mathbf{v}}_A\end{array}\right]$ and ${\mathbf{t}}_A=\left[\begin{array}{c}\omega \\ {\mathbf{v}}_A\end{array}\right]$ $\mathbf{t}$: instantaneous velocity
${\mathbf{v}}_P={\mathbf{v}}_A+(\mathbf{a}-\mathbf{p})\times \omega$
so, ${\mathbf{t}}_P=U{\mathbf{t}}_A$ , where $U=\left[\begin{array}{cc}I& 0\\ A-P& I\end{array}\right]$

computation of the twist parameter

* centriod $c=\frac{1}{3}{\sum }_{i=1}^3{p}_i$ likewise : $\dot{c}=\frac{1}{3}{\sum }_{i=1}^3{\dot{p}}_i$
$\mathbf{P}=[p_1-c,p_2-c,p_3-c]$
$\dot{\mathbf{P}}=[{\dot{p}}_1-\dot{c},{\dot{p}}_2-\dot{c},{\dot{p}}_3-\dot{c}]$
$\dot{\mathbf{P}}=\Omega \mathbf{P}$

$D\cdot vect(\Omega )=vect(\dot{\mathbf{P}})$, where $D=\frac{1}{2}[tr(P)I-P]$
we can use the equation to find angle velocity $\omega$:

1. find D: $D=\frac{1}{2}[tr(P)I-P]$
2. find: $vert(\dot{P})$
3. use: $D\cdot vect(\Omega )=vect(\dot{\mathbf{P}})$ and compare both side ,then find $\omega$

compatibility conditions of velocity
method1: check all points use $(\dot{\mathbf{p}}-\dot{\mathbf{a}})(\mathbf{p}-\mathbf{a})=0$
method2:

• position
  $P^{T}P=C$
• velocity
  ${\dot{P}}^{T}P+P^T\dot{P}=0$
  i.e. ${\dot{P}}^{T}P$ is skew-symmetric matrix

compatibility conditions of accelerations:
$\because \ddot{\mathbf{p}}=\ddot{\mathbf{a}}+\dot{\Omega }(\mathbf{p}-\mathbf{a})+\Omega (\dot{\mathbf{p}}-\dot{\mathbf{a}})$
$\therefore \ddot{\mathbf{p}}=\ddot{\mathbf{a}}+(\dot{\Omega }+{\Omega }^2)(\mathbf{p}-\mathbf{a})$
angular-acceleration matrix:
$W=\dot{\Omega }+{\Omega }^2$
$vect(W)=vect(\dot{\Omega })=\dot{\omega }$
$tr(W)=tr({\Omega }^2)=-2||\omega |{|}^2$
$\dot{t}=\left[\begin{array}{c}\dot{\omega }\\ \dot{\mathbf{v}}\end{array}\right]$

computation of angular-acceleration from piont-acceleration:
$\ddot{\mathbf{c}}=\frac{1}{3}{\sum }_{i=1}^3\ddot{{\mathbf{p}}_{\mathbf{i}}}$
$\ddot{P}=\left[\begin{array}{ccc}\ddot{{\mathbf{p}}_1}-\ddot{\mathbf{c}}& \ddot{{\mathbf{p}}_2}-\ddot{\mathbf{c}}& \ddot{{\mathbf{p}}_3}-\ddot{\mathbf{c}}\end{array}\right]$
$\ddot{\mathbf{p}}=(\dot{\Omega }+{\Omega }^2)P$
$D\dot{(}\omega )=vect(\ddot{P}-|Omeg{a}^{2}P)$

verify the angle acceleration compatible

Lecture5 Kinematics and Statics of Robotic Mechanical Systems

basic definitions

1. kinematic chain: llinks + kinematic pairs
   1. upper kinematic pairs: point,line contact.
   2. lower kinematic pairs: surface contact
2. type of lower kinematic pairs
   1. R: rotating pair, or revolute joints
   2. P: sliding pair, or prismatic joints
3. first links: manipulator base
4. last links: end effector(EE)

Denavit-Hartenberg table

DH parameters

1. $z_i$: ith axis
2. $x_i$: is defined as the common perpendicular to $z_{i-1}$ and $z_i$ from $z_{i-1}$ to $z_i$.
3. $y_i$: $\mathbf{y}=\mathbf{z}\times \mathbf{x}$, right hand rule
4. $a_i$: distance between $z_i$ and $z_{i+1}$
5. $b_i$: intersection of $x_{i+1}$ and $z_i$ , O’ in $z_i$
6. ${\alpha }_i$: the angle of $z_i$ and $z_{i+1}$ (from $z_i$ to $z_{i+1}$ along $x_{i+1}$)
7. ${\theta }_i$: the angle of $x_i$ to $x_{i+1}$ , (from $x_i$ to $x_{i+1}$ , along $z_i$)
8. $x_1$: position arbitrary, fixed and parallel to floor
9. link from o to n, frame from 1 to n+1

---

$$[Q_i{]}_i=\left[\begin{array}{ccc}\cos {\theta }_i& -\cos {\alpha }_i\sin {\theta }_i& \sin {\alpha }_i\sin {\theta }_i\\ \sin {\theta }_i& \cos {\alpha }_i\cos {\theta }_i& -\sin {\alpha }_i\cos {\theta }_i\\ 0& \sin {\alpha }_i& \cos {\alpha }_i\end{array}\right]$$
$$[a_i{]}_i=\left[\begin{array}{c}{a}_i\cos {\theta }_i\\ a_i\sin {\theta }_i\\ b_i\end{array}\right][b_i{]}_i=\left[\begin{array}{c}{a}_i\\ b_i\sin {\alpha }_i\\ b_i\cos {\alpha }_i\end{array}\right]$$
$$a_i=Q_i{b}_i$$

general IKP for six_revolute manipulator

• the EE position and orientation:
  • orientation :
    $Q_1{Q}_2{Q}_3{Q}_4{Q}_5{Q}_6=Q$
  • position:
    $P=a_1+Q_1{a}_2+Q_1{Q}_2{a}_3+Q_1{Q}_2{Q}_3{a}_4+Q_1{Q}_2{Q}_3{Q}_4{a}_5+Q_1{Q}_2{Q}_3{Q}_4{Q}_5{a}_6$

Lecture6 Innovative Design and Application

6.1 parallel manipulators

1. types of manipulators
   $$manipulator\{\begin{array}{l}SerialMainpulator\\ ParallelManipulator\end{array}$$
2. basic concepts and definitions
   • Parallel mechanism: close-loop mechanism, mobile platform is connected to the base by at last two independent kinematic chains
   • Similar terms: closed-loop mechanism, parallel robot, parallel manipulator
   • Antonyms: open-loop mechanism, serial manipulator, serial robot
   • Parallel Kinematic Machine(PKM): machine tool based on a parallel mechanism.
3. advantages and disadvantages of PKM
   • advantages
     • high rigidity(刚度) and stiffness(强度)
     • high speed
     • high accuracy
     • high flexibility
   • disadvantages
     • small workspace and cutting volume
     • difficulty to design
     • coupled control system

6.2 innovation design(parallezation of serial robots)

synthesis starting from the actuating structure

synthesis of parallel links based on DOF constaints

requirements on parallel actuated arms

6.3 reconfigurable robotic manipulators

6.4 modular flexure joints for compliant mechanisms

PZT(压电陶瓷) actuator

6.5applications

Lecture7