Rob 501 - Mathematics for Robotics
HW #1
Prof. Grizzle
Due 3 PM on Thur., Sept 14, 2017
We are using Gradescope.
Remarks: The initial lectures in Rob 501 are not required for working this HW set. These problems should
be doable using standard undergraduat
ROB 501: Mathematics for Robotics
Instructor: Prof. Jessy W. Grizzle, 4421 EECS Bldg., [email protected], 734-7633598
Class Periods: Lecture meets T-Th, 10:30 to 12:00, in 1500 EECS. Lectures will be
recorded and posted to Canvas.
Due to the extra-large c
Movie Segment
Locomotion Gates with Polypod,
Mark Yim, Stanford University,
ICRA 1994 video proceedings
Jacobian
Differential Motion
Linear & Angular Motion
Instantaneous
Velocity Propagation
Kinematics
Explicit Form
Static Forces
Differential Motion
Introduction to Robotics (CS223A)
Homework #2 Solution
(Winter 2007/2008)
1. The following sketch represents a generic open, serial, kinematic-chain.
Here each kinematic joint connects two adjacent members. Assume that the relative
displacement between ad
Movie Segment
The Hummingbird, IBM Watson
Research Center, ICRA 1992
video proceedings
Manipulator Kinematics
Manipulator
Link i -1
Link Description
Joint axis i -1
Denavit-Hartenberg Notation
Frame Attachment
Forward Kinematics
Axis i
Link Descriptio
Spatial Descriptions
Task Description
Kinematics
Transformations
Representations
Configuration Parameters
Manipulator
Prismatic Joint
A set of position parameters that describes
the full configuration of the system.
Revolute
Joint
k
Lin
9 parameters/li
Introduction to Robotics (CS223A)
Homework #1 Solution
(Winter 2007/2008)
1. A frame cfw_B and a frame cfw_A are initially coincident. Frame cfw_B is rotated about
YB by an angle , and then rotated about the new ZB by an angle . Determine
the 3 3 rotation
Introduction to Robotics (CS223A)
(Winter 2007/2008)
Homework #2
Due: Wednesday, January 30
1. The following sketch represents a generic open, serial, kinematic-chain.
Here each kinematic joint connects two adjacent members. Assume that the relative displ
Introduction to Robotics (CS223A)
Homework #3 Solution
(Winter 2007/2008)
1. You are given that a certain RPR manipulator has the following transformation
matrices, where cfw_E is the frame of the end effector.
0
1T
=
c1 s1
s1
c1
0
0
0
0
0
0
1
0
0
ET
=
0
Introduction to Robotics (CS223A)
(Winter 2007/2008)
Homework #3
Due: Wednesday, February 06
1. You are given that a certain RPR manipulator has the following transformation matrices,
where cfw_E is the frame of the end effector.
0
1T
=
c1 s1
s1
c1
0
0
0
ROB 501: Mathematics for Robotics
Instructor: Prof. Jessy W. Grizzle, 4421 EECS Bldg., [email protected], 734-7633598
Class Periods: Lecture meets T-Th, 10:30 to 12:00, in 1200 EECS. Lectures will be
recorded and posted to Canvas.
Due to the extra-large c
Rob 501 Fall 2014
Lecture 24
Newton-Raphson & Contraction Mapping
Typeset by: Kevin Chen
Proofread by: Yong Xiao
h
n
Let h : Rn Rn , h
x (x) exists at each point, x R , and moreover, x (x) is
a continuous function. One says h is C 0 , derivative exits and
Rob 501 Lecture note 23
Typed by Ilsun Song and Proof-read by Yunxiang Xu
Sequence
Def A set of vectors indexed by the non-negative integers is called a sequence
(Xn ) or cfw_Xn . Let (X) be a sequence and n1 < n2 < n3 . be an infinite set of
strictly inc
Rob 501 Fall 2014
Lecture 25
Typeset by: Yunxiang Xu
Proofread by: put name here
Def.
A set C is bounded if < such that C Br (0).
Bolzano-Weierstrass Theorem (Sequential Compactness Theorem): In
a nite dimensional normed space (X , R, | |), the following
Rob 501 Fall 2014
Lecture: Random Vector
Typeset by: Xianan Huang
Proofread by: Josh Mangelson
1
Probability Review
Given: (, F , P ) a probability space
X : R random variable
2
Random Vector
Def. A random vector X : RP where each component
x1
x2
.
.
, x
Rob 501 Fall 2014
Lecture 20
Typeset by: Jeff Koller
Proofread by: Yevgeniy Yesilevskiy
The beginning of this lecture completes the derivation of the Kalman Filter.
The material covered can be found on pages 5-7 of the KalmanFilterDerivationUsingCondition
ROB 501 - Lecture December 4th
Vittorio Bichucher, Mia Stevens
December 20, 2014
1
Additional Facts
8
All norms | | : X [0, ) are convex (proof with triangle inequality)
8
For all 1 < , | | is convex. Hence, on Rn :
ni=1 = |xi |3
(1)
is convex.
8
Let r
ROB501 Class Notes 11.11.14
Yevgeniy Yesilevskiy
November 26, 2014
1
Conditional Mean or Expectation
Z
(x2 ) = Ecfw_X1 |X2 = x2 =
Z
x1 f (x1 |x2 ) dx1 =
Rn
Rn
= argminz=g(x2 ) Ecfw_kX1 zk2 |X2 = x2
Theorem: Let x
where g varies over all functions g : R
Rob 501 Fall 2014
Lecture 16
Typeset by: Kurt Lundeen
Proofread by: Connie Qiu
QR Decomposition or Factorization: Let A be a real m n matrix with
linearly independent columns (rank of A = # columns). Then there exists an
m n matrix Q with orthonormal colu
Rob 501 Fall 2014
Lecture 13
Typeset by: Ming-Yuan Yu
Proofread by: Ilsun Song
1. Normal Equations, Take 2: Weighted Least Squares
Let Q be an n n positive definite matrix
Q>0
and let the inner product on Rn be
hx, yi = xT Qy
We re-do A = b:
A = n m, n m,
Rob 501 Fall 2014
SVD Lecture
Typeset by: Joshua Mangelson
Proofread by: Katie Skinner
We will use the SVD (Singular Value Decomposition) to understand "numerical
rank" of a matrix, "numerical linear independence", etc.
Def. Rectangular diagonal matrix:
=
Rob 501 Handouts
Linear Algebra and Geometry
J.W. Grizzle
Fall 2015
Compiled on Friday 11th September, 2015 at 10:37
1
Linear Algebra by Gabriel Nagy
You can download the full textbook at the link given on the next
page. We have permission to use it. Prof
Rob 501 Fall 2014
Lecture 11
Typeset by: Su-Yang Shieh
Proofread by: Zhiyuan Zuo
Symmetric Matrix
Let A = n n real matrix.
Def.: A = AT is a symmetric matrix.
Claim 1 The eigenvalues of the symmetric matrix are real.
Proof: Let C be an eigenvalue. To show
Rob 501 Fall 2014
Lecture 12
Typeset by: Yong Xiao
Proofread by: Pedro Donato
Review:
A square matrix O is orthogonal if Oo pO = I columns of O are or-
thonormal.
>
x Mx = x
>
M +M >
2
x, where
M +M >
2
is the symmetric part of M .
P is a real symmetri
Rob 501 Fall 2014
Lecture 14
Typeset by: Bo Lin
Proofread by: Hiroshi Yamasaki
Weighted Least Square
< x, y >= xT Qy, Q > 0
Overdetermined equations
Ax = b, x Rn , b Rm , n < m, rank(A) = n.
x = [AT QA]1 AT Qb.(QAx = Qb, AT QAx = AT Qb, x = [AT QA]1 AT Qb
Rob 501 Fall 2014
Lecture 05
Typeset by: Meghan Richey
Proofread by: Su-Yang Shieh
Thm. Let (X, F ) be an n-dimensional vector space (n finite). Then any set
of n linearly independent vectors cfw_v 1 .v n is a basis.
Proof (Handout). To show x X = x Span
Rob 501 Fall 2014
Lecture 04
Typeset by: Xiangyu Ni
Proofread by: Sulbin Park
Def. Let (X , F) be a vector space. A linear combination is a finite sum of the
form 1 x1 + 2 x2 + + n xn where n 1, i F, xi X .
x1 i
x i
2
i
Remark: x = . , where xi means in
Rob 501 Fall 2014
Lecture 02
Typeset by: Ross Hartley
Proofread by: Jimmy Amin
Second Principle of Induction (Strong Induction)
Let P(n) be a statement about the natural numbers with the following properties:
1. Base Case: P (1) is true
2. Induction: If P
Rob 501 Fall 2014
Lecture 07
Typeset by: Zhiyuan Zuo
Proofread by: Vittorio Bichucher
Elementary Properties of Matrices (Assume Known)
A = n m matrices with coefficient in R or C.
Define rank of A = # of linearly independent columns of A.
Thm. rank(A) = r