Solutions Manual to Introduction to Robotics Mechanics and Control-Pearson

SOLUTIONS MANUAL INTRODUCTION TO ROBOTICS MECHANICS AND CONTROL Chapter 1 Introduction Exercises 1.1) Here's just an exarnple of а rcaso11aЫc response: (ref. [8] in Chap. 1) Denavit & Harte11Ьcrg developed methodology fог descriЬing linkages. George Devol patents design of rst roЬoL First uniшate roЬot installed. Shakey RoЬot deve1oped at S.R.I. RoЬot institute of America formcd. Unimation Ьecomes rst RoЬot Со. to Ье protaЫe. First Puma RoЬot shipped to GM. Total U.S. markct cxceeds 500 million dollars (annt1al revenue). 1955 1 %1 1961 1968 1975 1975 1978 1985 Developments might Ье split into а 1echnica] list апd а business list. 1.2) (Based оп 1981 numЬers) ■ Source: L. Conigliaro, "roЬotics presentation, institutional investors conr.'·, Мау 28, 1981. Bache Ne,vs1etter 81-249.: 1.3) People Are RelliЫe, But More ExJJenslve Every Year $ВО U.S. Autonюtivc $51 $40 130 ш 111 so '"' 1111 1114 1181 1111 1115 Other - 32.2% Pallctiic/Packaging - 2.8% 11 ■ ■ - Machining - 6.8%, Material Handling - 11.8% [] ЛssеП1Ьlу - 22.4% Welding - 23.9% 1.4) Kim!matics is the study of position and derivatives of position without rcgard to forces which causc the motion. �Vork.�pace is the locus of positions a11d orieпtations nchievaЬle Ьу the end-cГfector of а manipulator. Trajectory is а timc bascd functioп \Vhich specifies the position (and higher derivatives) of the robo[ mechaпism for апу ,.:alue of time. 1.5) F,·ame is а coordinatc systcrn, usually spccified in position and orientation relative to some imbedding frame. Degrees о/ freedom is the number of indcpendcnt variaЫcs which must Ьс specificd iп oгder to coшpletely locale all memЬers of а (rigid-Ьody) mechanism. Positio,i control implies the use of а control system, usually in а closcdloop manner, to control thc position of one ог more moving bodies. 1.6) Force co11trol is the use of (usually closed-loop) aJgorHhms to control the forces of contac\ generated ,vhcп а roЬot touches its work environment. А гоЬоt prog,шnmi,ig Janguage i� а programming ]anguagc intcnded for use i[l specifyiпg шапiрulator actions. 1.7) Structura/ stiffness is tl1c "К" in F = КЛХ (А.К.А "Hooke's law") which descriЬes the rigidity of some structure. Nonlinear con/rol refers to а closed loop control system in which either the systcm to Ье controlled, or the control algorithm itself is nonJinear in nature. Ojf line programming is thc proccss of crcating а program for а dcvicc \Vithout access to that device. 1.8) See references. For example, in 1985 average labor costs of $ 15 to $20 per hour are reasonable (depending how fringe benefiLs are calculated). 1.9) Obviously it has increased dramatically. Recently (1988-1990) the ratio douЫes or even triples eoch year. 1.10) See Figurc 1.3, but use Jatcst dara you сап fi11d. Chapter 2 Spatial Descriptions and Transformations Exercises 2.1) R = rot(x, ф) rot(z, 0) 1 = [О О 2.2) R о Сф Sф = rot(i, 45 °) rot(y, 30° ) .866 = [ .353 -.353 О .707 .707 .5 -.612 ] .612 2.3) Since rotations are performed aЬout axes of the frame being rotated, these are Euler-Angle style rotations: R = rot(z, 0) rot(x, ф) We might з]sо use the following reasoning: = [rot(i, -Ф) rot(z, -0)]-1 = rог 1 (z, -0) rог 1 (х, -ф) = rot(z, 0) rot(i, ф) Yet another way of viewing the same operation: 1st rotate Ьу rot(z, 0) 2nd rotate Ьу rot(z, 0) rot(i, ф) roГ 1 (z. 0) 2.3) (co111inued) (This is а similarity transform) Composing these two rotations: = rot(z, 0) rot(x, ф) гог1 (z, 0) • rot(z. 0) = rot(i. 0) rot(x, ф) = = si [со s; [со -S0 О ' со о ][о О I О о Сф Sф S0Sф] -S0Сф С0Сф -С0Sф Sф Сф -�Ф] Сф 2.4) This is the same as 2.3 опlу with numbers. R 2.5) = rot(i, 30° ) rot(i'. 45) .866 -.353 = [ ,50 ,612 О .707 .353 -,612 ] .707 lf V; is an eigenvector of R, then RV; = 7V; I.f the eigenvalue associated with V; is 1, then RV, = V, Hence the vector is поt changed Ьу the rotation R. So V1 is the axis of rotation. 2.6) lmagine а frame {А} whosc Z axis is aligncd with the direction ic: Uzt\U) Then, the rotation with rotates vcctors аЬош k Ьу 0 degrees could Ье written: \ х, We write the description of (А) in (U) as: �R = [АВ С D Е F Кх К, ] К, lf we multiply out Eq. [ 1) above, and then simplify using А 2 + 8 2 + С2 = 1, D2 + Е2 + F2 = l, [АВС) [DEF] = О, [ABC)@[DEFJ = [К,К,.К,] we arrive at Eq. (2.80) in the Ьооk. Also, see [R. Paul]' page 25. у \/!AI \л, l'z К 2.7) R12 R22 R,2 Rн R23 ] Rзз Lel R11 R = [ R21 R,1 (1) Compute R11 + R,, + Rз, = N (2) N-1 lfN=З,thenO=Acos ( - -) =0". Since 2 rotation is zero, К is arblLrary. (3) lf N = -1, then В= Acos(-1) = 180'. 1n this case Eq. (2.80) becomes: 2К 1 -1 rot U Ул Now: 2.24) This is ablttricky hcrc's mostofit: "Р = �R 8 Р doesn't change length, so :. Ар. Ар _ Вр . В р =0 ,_____, ,_____, = о А (' р - в Р). ( р+ в Р) F <; :. F .L G 8 Р = (;R + 1)-'с (You can show ;R + 1 non-sha,var) :. F = (1R - l)(;R + /)-1 G so, F = BG (Now, one can show that if Х •У= О and if Х = АУ then а is skew-symmetric) :. В Е skew-sym matrices (/ - B)tR 2.25) = 1 + В (now can show (1-В) non-show) Def. of e-va\: л; V; = R V; from physical insight we know V; = R V; v.1 hen Vi is the axis of rota� tion, .·.1 Л 1 = 1 1 from (2.80) one сап show (with some work) that det (R) = \. From linear algebra we have: �;л; = trace(r) ,r;л; = det(r) :. л1л2лз = 1 or л2л3 =1 [IJ And compute the trace(r) from (2.80) to get л 1 + л2 + л3 = 1 + 2 cos 0 or л, + лз = 2 cos О Now solve [!) & [2] above for л2 and л3 to get л,+ л2 ..!.. л, =2соs0; J.j-2cos01.2+l=0 = cose ± Jcos2 0 - 1 А2 = cos0 ±i sin0 :. ,., = 1 А2 = cos0 + i sinB = е18 Л з = cosB -i sinB = e-ie Q.E.D. [2] 2.26) SomeЬody ptease send me а !iiшple. proof of tl1is. (For :::iny given Euler convcntion it is not hard.) g -1 0 2.27) А8Т = [ ;т= [-r 2.29) св Т = -� 2.28) 2.30) сА Т = 2.з1J 2.32) ;т = �т = о о о о о 1 о о �] -1 о о о о 0.5 -0.866 -0.866 -0.5 о о о о [ о о -1 о о о -0.5 0.866 0 866 -0.5 () о [ о [ g ] - о о о -1 о о - о] о о 2.34) �т = 2 3 • 0.5 ] -3•?-866 0.866 -i] о о -1 о о о о -0.866 -0.5 о �] i] 0.5 0.866 -0.866 [0-� в [ о 2.33) с Т = -0i �] 0.866 -0.5 0.866 0.5 +1 о �] 0.866 о -3'.86+2 0.5 -0.866 о -4'.8 - 1.5] [0-� � о -1 о о 2.35) Any R сап Ье given: R = Rx(a)Ry(IJ)Rz(y) and det(R) = det(Rx(a)) • det(Rr(/i)) • dct(Rz(y)) Using thc formulas for rotation aЬout а principlc axis ((2.77) - (2.79)) its easy to show det(R,(a)) = 1 Va =I vp dct(R,(y)) = 1 Vy del(R,(/i)J : . dct(R) 2.36) = 1 • 1 • 1 = 1 Q.E.D. From Cayley's formula, thc numЬcr of parame­ ters needed to specify а rotation is the number of free parameters i11 an NXN skew symmctric matrix, which is ½ epsilon? Yes ,1 = (r23 - r32)/4s1 ,,. =0 s; = -1 /2(r22 + r,з) •2 = (rз, - r13)/4e,. е� > cpsilon? Ез = (r12 - r21)/4E1 Е)·=Й Yes Е\ &2 =Ff = r12/2E1 Ез = r1з/2Е1 ,, = о •l= t(l-,,,J sf > epsilon? Yes t2 = f,j Ез = r2з/2s2 2.40) SimiJar to зlgorithms of Section 2.8. 2.41) Similar to algorithms of Section 2.8. 1 No ,, = о Е3:::)