CPSC 452 Inverse Kinematics Prof. Oussama Khatib, Stanford

CPSC 452 Inverse Kinematics Prof. Oussama Khatib, Stanford

CPSC 452 Inverse Kinematics Prof. Oussama Khatib, Stanford University Prof. Dezhen Song, Texas A&M University

X Workspace

Reachable Workspace Dexterous Workspace X

Given X Find q=(1, 2, 3)

Algebraic Solution The kinematics of the example seen before are: c123 s

123 B 0

T T

W 3 0

0 s123 0 l1c1 l2 c12

c123 0 l1s1 l2 s12 0 1

0 0 0

1 Assume goal point is specified by 3 numbers:

c s B

WT 0

0 s c

0 0 0 x

0 y 1 0 0 1

Algebraic Solution (cont.) By comparison, we get the four equations: c c123

s s123 x l1c1 l2 c12

y l1s1 l2 s12 Summing the square of the last 2 equations: x 2 y 2 l12 l22 2l1l2 c2

From here we get an expression for c2 2

2 2 1

x y l l c2 2l1l2

2 2

Algebraic Solution (III) When does a solution exist? What is the physical meaning if no solution exists? Two solutions for 2 are possible. Why?

Using c12=c1c2-s1s2 and s12= c1s2-c2s1: x k1c1 k 2 s1 y k1s1 k 2 c1

where k1=l1+l2c2 and k2=l2s2. To solve these eqs, set r=+ k12+k22 and =Atan2(k2,k1). Algebraic Solution (IV)

k1

l2 2

k2 l1

Then: k1=r cos , k1=r sin , and we can write: x/r= cos cos 1 - sin sin 1 y/r= cos cos 1 - sin sin 1 or: cos(+1) = x/r, sin(+1) =y/r

Algebraic Solution (IV) Therefore: +1 = Atan2(y/r,x/r) = Atan2(y,x)

and so: 1 = Atan2(y,x) - Atan2(k2,k1) Finally, 3 can be solved from: 1+ 2+ 3 =

Geometric Solution IDEA: Decompose spatial geometry into several plane geometry problems

y L2 L1 x

Applying the law of cosines: x2+y2=l12+l22 - 2l1l2cos(180+2)

Recently Viewed Presentations

  • Physical Science 18.2 The Electromagnetic Spectrum Reading Guide

    Physical Science 18.2 The Electromagnetic Spectrum Reading Guide

    What does RADAR stand for? Contrast the wavelength and frequencies radio waves and infrared rays. List the colors of the visible light spectrum from longest to shortest wavelength. List two applications UV rays have in health and medicine. Contrast the...
  • REDCap Basics

    REDCap Basics

    Event 3 - Month 1 Instrument for Event 3 and Data was listed form previous event or the default for yes is enabled . Add the choice value with underscore before the equal sign and variable name in single quotes...
  • Fall 2004, CIS, Temple University CIS527: Data Warehousing,

    Fall 2004, CIS, Temple University CIS527: Data Warehousing,

    Mining Complex Types of Data Multidimensional analysis and descriptive mining of complex data objects Mining spatial databases Mining multimedia databases Mining time-series and sequence data Mining text databases Mining the World-Wide Web Summary Similarity Search in Multimedia Data Description-based retrieval...
  • World History - Jenks Public Schools

    World History - Jenks Public Schools

    World History The Early Beginnings of Mankind ... Know Prehistory Anthropology Archaeology Paleontology Artifacts Hominid Radiocarbon Dating Used to determine the age of artifacts and fossils "Lucy" was unearthed in 1974 by Donald Johanson Radiocarbon dated to 3.2 million years...
  • The Krebs Cycle

    The Krebs Cycle

    The Krebs Cycle Occurs in the matrix of the mitochondrion Aerobic phase (requires oxygen) 2-carbon acetyl CoA joins with a 4-carbon compound to form a 6- carbon compound called Citric acid
  • 2016 Loyalty & Savings Book February 22 through

    2016 Loyalty & Savings Book February 22 through

    2016 LOYALTY & SAVINGS PROGRAM. February 22 — June 24, 2016 . Official Promotion . Rules. The 2016 Colony Foods Loyalty and Customer Appreciation Program offer the opportunity to earn Retailer or Colony Foods gift cards or credit towards a...
  • Language and Identity

    Language and Identity

    Purpose of Lang & Id. For this topic we are concerned with how an individual identity is constructed through language. Besides . the human desire to belong to a community, exists the desire to be a . unique
  • 4 - materlakes.org

    4 - materlakes.org

    Photomicrograph: Skeletal muscle (approx. 460x). Notice the obvious banding pattern and the fact that these large cells are multinucleate. Nuclei Striations Part of muscle fiber (cell) Figure 4.10b (b) Cardiac muscle Description: Branching, striated, generally uninucleate cells that interdigitate at...