Fundamentals of Robotic Mechanical Systems – Fourth Edition | Jorge Angeles | McGill University | Solutions to Selected Problems

FUNDAMENTALS OF

ROBOTIC MECHANICAL SYSTEMS

Theory, Methods, and Algorithms
Fourth Edition


Solutions to Selected Problems



Jorge Angeles




Department of Mechanical Engineering &
Centre for Intelligent Machines (CIM)
McGill University
Montreal, Quebec, Canada





December 2015
c⃝Jorge Angeles

,1 An Overview of Robotic Mechanical Systems
1.1 Machine: A historical account

Here is an account of the definitions of machine, taken from ( D u diţ ă and Diaconescu, 1987):
Different definitions of machine have been given by scholars for more than two millennia, starting with
Vitruvius in 28 B.C., namely,

• A machine is a combination (system, assemblage) of moving material bodies (Vitruvius, 28 B.C.;
Hachette, 1811; Borgnis, 1818; Beck, 1859; Reuleaux, 1875; Koenigs, 1901)
• A machine is generally composed of three parts: a motor part, a transmission part, and an exe-
cution part (Euler, 1753; Bogolyubov, 1976)
• A machine produces mechanical work, or performs productive operations, actions, or effects (Vit-
ruvius, 28 B.C.; Poncelet, 1824; Reuleaux, 1900; Koenigs, 1901; Bogolyubov, 1976)
• A machine transforms or transmits forces (Vitruvius, 28 B.C.; Leupold, 1724; Euler, 1753; Bo-
golyubov, 1976; Reuleaux, 1900; Koenigs, 1901)
• A machine is characterized by deterministic motions (Hachette, 1811; Leupold, 1724; Reuleaux,
1875; Borgnis, 1818; Reuleaux, 1900)
• A machine is an artifact (Leupold, 1724)

Beck, Th., 1875, Beiträge zur Geschichte des Maschinenbaues, J. Springer, Berlin.
Bogolyubov, A. N., 1976, Teoriya mekhanismov v istoricheskom razvitii (Theory of Mechanisms and
its Historical Development), Nauka, Moscow (in Russian).
Borgnis, G. A., 1818, Trait´e Complet de M´ecanique Appliqu´ee aux Arts. Trait´e des Compositions des
Machines, Paris.
Du d iţ ă , Fl. and Diaconescu, D., 1987, Optimizarea Structurală a Mecanismelor (Optimization of Me-
chanisms ), in Romanian, Ed. Tehnicá (Publishers), Bucharest.
Hachette, 1811, Trait´e Élémentaire des Machines, Paris.
Koenigs, F., 1901, “Etude critique sur la th´eorie g´en´erale des m´ecanismes,” Comptes Rendus de
l’Acad´emie des Sciences, Vol. 133.
Leupold, J., 1724, Theatrum Machinarium Generale, Leipzig.
Poncelet, J. V., 1824, Trait´e de M´ecanique Appliqu´ee aux Machines, Li`ege.
Reuleaux, F., 1875, Theoretische Kinematik, Braunschweig.
Reuleaux, F., 1900, Lehrbuch der Kinematik, Braunschweig.
Vitruvius, P. M., 28 B.C., De Architectura, Libri X.

1.3 Definitions for “machine,” “mechanism,” and “linkage”:

Machine

• Definitions in Merriam Webster’s Collegiate Dictionary (on-line, 2002):
– (archaic): a constructed thing whether material or immaterial.
– an assemblage of parts that transmit forces, motion, and energy one to another in a prede-
termined manner
– an instrument (as a lever) designed to transmit or modify the application of power, force, or
motion


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, – a mechanically, electrically, or electronically operated device for performing a task (a calcu-
lating machine, a card-sorting machine)
Comment: comprehensive definitions when considered as a whole
• An apparatus for transformation of power, materials, and information to substitute or simplify
physical or intellectual work (Frolov, 1987). Comment: a comprehensive definition, that includes
computers
• Mechanical system that performs a specific task, such as the forming of material, and the trans-
ference and transformation of motion and force, Vol. 38, Nos. 7–10 (2003) of Mechanism and
Machine Theory on Standardization of Terminology. Comment: leaves computers out
• An apparatus for applying mechanical power, having several parts, each with definite function (The
Concise Oxford Dictionary). Comment: same as above
• An apparatus consisting of interrelated parts with separate functions, used in the performance of
some kind of work (The Random House College Dictionary). Comment: ditto
• Any system in which a specific correspondence exists between an input form of energy or in-
formation and the corresponding ones at the output (Loosely translated from Le Petit Robert).
Comment: as comprehensive as Frolov’s

Mechanism

• A piece of machinery (Merriam Webster’s Collegiate Dictionary (on-line, 2002)). Comment: too
vague
• Definitions in Vol. 38, Nos. 7–10 (2003) of Mechanism and Machine Theory on Standardization
of Terminology.
– System of bodies designed to convert motions of, and forces on, one or several bodies into
constrained motions of, and forces on, other bodies. Comment: English could be terser, but idea
is fine.
– Kinematic chain with one of its components (link or joint) connected to the frame. Comment:
confuses mechanism with its representation as a kinematic chain
• Structure, adaptation of parts of machine; system of mutually adapted parts working together
(as) in machine (The Concise Oxford Dictionary)
• An assembly of moving parts performing a complete functional motion (The Random House
College Dictionary)
• A combination layout of pieces or elements, assembled with the goal of (producing) an operation
as a unit (Loosely translated from Le Petit Robert)
Comment: In all above definitions, the concept of goal or task is present
Linkage
• Definitions in Merriam Webster’s Collegiate Dictionary (on-line, 2002):
– a system of links. Comment: concise and comprehensive
– a system of links or bars which are jointed together and more or less constrained by having a
link or links fixed and by means of which straight or nearly straight lines or other point paths
may be traced. Comment: unnecessarily cumbersome and limited to path-generating linkages
• Kinematic chain whose joints are equivalent to lower pairs only (Vol. 38, Nos. 7–10 (2003) of
Mechanism and Machine Theory on Standardization of Terminology). Comment: confuses linkage
with its representation

The Concise Oxford Dictionary of Current English, 1995, Clarendon Press, Oxford


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, Frolov, K. V., (editor), 1987, Teoriya Mechanismov i Mashin (Theory of Mechanisms and Machines),
Vyschaya Shkola, Moscow (in Russian)
Mechanism and Machine Theory on Standardization of Terminology, 2003, Vol. 38, Nos. 7–10
Le Petit Robert, 1994, Dictionnaires Le Robert, Paris
Random House Webster’s College Dictionary, 1997, Random House, New York
Merriam Webster’s Collegiate Dictionary, on-line 2002

1.9 Here we want to estimate the time required to multiply two floating-point numbers. Since this time is
very small, it is more suitable to perform a large number of multiplications, say 107, which will require
a total time of the order of seconds.

• The Microsoft Visual C++6.0 program below was run on a Pentium IV 2.0:


#include <iostream>
#include <time.h>
int main()
{
clock_t start, end;
start =clock();
float a, i, CLOCKS_PER_mSEC=CLOCKS_PER_SEC/1000;
for(i = 1; i <= 10000000; i++)
a = 5 * 5;
std::cout << ‘‘a= ‘‘ << a << std::endl;
end = clock();
long duration=(long)(end-start)/CLOCKS_PER_mSEC;
std::cout << ‘‘Time: ‘‘ << duration<<’’ms’’<<std::endl;
return 0;
}
A Pentium IV 2.0 processor required 125 ms to perform the 107 multiplications, which amounts
to 1.25 × 10−8 s/mult, or 8 × 107 mult/s.
• The C program below was run for the same purpose on the CLUMEQ (Consortium Laval UQAM
McGill and Eastern Quebec for High Performance Computing) supercomputer, AMD Athlon
1900+ cluster:
# include "stdio.h"
# include "math.h"
# include <sys/time.h>
# include <sys/resource.h>
# define RUSAGE_SELF 0
\* calling process *\
main()
{
int getrusage(int who, struct rusage *rusage);
long diffsec, diffmsec;
float a;
int i;
struct rusage time_begin, time_end;
struct rusage *pb, *pe;
pb=&time_begin;


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