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Mech541 Kinematic Synthesis MECH541 KINEMATIC SYNTHESIS Lecture Notes Jorge Angeles Department of Mechanical Engineering & Centre for Intelligent Machines McGill University, Montreal (Quebec), Canada Shaoping Bai Department of Mechanical Engineering Aalborg University Aalborg, Denmark c January 2016 These lecture notes are not as yet in final form. Please report corrections & suggestions to Prof. J. Angeles Department of Mechanical Engineering & McGill Centre for Intelligent Machines McGill University 817 Sherbrooke St. W. Montreal, Quebec CANADA H3A 0C3 FAX: (514) 398-7348 [email protected] Contents 1 Introduction to Kinematic Synthesis 5 1.1 The Role of Kinematic Synthesis in Mechanical Design . 5 1.2 Glossary..................................... 9 1.3 Kinematic Analysis vs. Kinematic Synthesis . 12 1.3.1 ASummaryofSystemsofAlgebraicEquations . 14 1.4 Algebraic and Computational Tools . 15 1.4.1 The Two-Dimensional Representation of the Cross Product ................................. 15 1.4.2 Algebra of 2 2Matrices ....................... 17 × 1.4.3 Algebra of 3 3Matrices ....................... 18 × 1.4.4 Linear-Equation Solving: Determined Systems . 18 1.4.5 Linear-Equation Solving: Overdetermined Systems . 22 1.5 Nonlinear-equation Solving: the Determined Case . .. 32 1.5.1 TheNewton-RaphsonMethod . 34 1.6 Overdetermined Nonlinear Systems of Equations . ... 35 1.6.1 TheNewton-GaussMethod . 35 1.7 SoftwareTools.................................. 38 1.7.1 ODA:MatlabandCcodeforOptimumDesign . 38 1.7.2 Packages Relevant to Linkage Synthesis . 40 2 The Qualitative Synthesis of Kinematic Chains 45 2.1 Notation..................................... 45 2.2 Background ................................... 46 2.3 KinematicPairs................................. 51 2.4 Graph Representation of Kinematic Chains . 54 2.5 GroupsofDisplacements ............................ 56 2.5.1 DisplacementSubgroups . 60 2.6 KinematicBonds ................................ 64 2.7 The Chebyshev-Gr¨ubler-Kutzbach-Herv´eFormula . ....... 66 2.7.1 Trivial Chains . 67 i 2.7.2 ExceptionalChains ........................... 71 2.7.3 ParadoxicalChains ........................... 74 2.8 ApplicationstoRobotics ............................ 74 2.8.1 The Synthesis of Robotic Architectures and Their Drives . 74 3 Function Generation 79 3.1 Introduction................................... 79 3.2 Input-OutputFunctions ............................ 80 3.2.1 PlanarFour-BarLinkages . 80 3.2.2 TheDenavit-HartenbergNotation. 83 3.2.3 SphericalFour-Bar-Linkages . 84 3.2.4 SpatialFour-Bar-Linkages . 89 3.3 ExactSynthesis ................................. 93 3.3.1 PlanarLinkages............................. 93 3.3.2 SphericalLinkages ........................... 98 3.3.3 SpatialLinkages.............................100 3.4 Analysis of the Synthesized Linkage . 101 3.4.1 PlanarLinkages.............................101 3.4.2 SphericalFour-BarLinkages . .112 3.4.3 SpatialFour-BarLinkages . .114 3.5 ApproximateSynthesis .............................121 3.5.1 The Approximate Synthesis of Planar Four-Bar Linkages . 124 3.5.2 The Approximate Synthesis of Spherical Linkages . 126 3.5.3 The Approximate Synthesis of Spatial Linkages . 127 3.6 LinkagePerformanceEvaluation. 131 3.6.1 Planar Linkages: Transmission Angle and Transmission Quality . 131 3.6.2 Spherical Linkages: Transmission Angle and Transmission Quality . 136 3.6.3 Spatial Linkages: Transmission Angle and Transmission Quality . 137 3.7 DesignErrorvs.StructuralError . 139 3.7.1 Minimizing the Structural Error . 142 3.8 Synthesis Under Mobility Constraints . 145 3.9 SynthesisofComplexLinkages. .146 3.9.1 SynthesisofStephensonLinkages . 146 4 Motion Generation 147 4.1 Introduction...................................147 4.2 PlanarFour-barLinkages. .. .. .147 4.2.1 DyadSynthesisforThreePoses . 149 4.2.2 DyadSynthesisforFourPoses . .149 4.2.3 DyadSynthesisforFivePoses . .152 ii 4.2.4 Case Study: Synthesis of a Landing Gear Mechanism . 153 4.2.5 The Presence of a P JointinDyadSynthesis . 159 4.2.6 ApproximateSynthesis . .164 4.3 SphericalFour-barLinkages . 169 4.3.1 DyadSynthesisforThreeAttitudes . 171 4.3.2 DyadSynthesisforFourAttitudes. 172 4.3.3 DyadSynthesisFiveAttitudes. 173 4.3.4 Spherical Dyads with a P Joint ....................174 4.3.5 ApproximateDyadSynthesis . .175 4.3.6 Examples ................................181 4.4 SpatialFour-barLinkages . .. .. .187 5 Path Generation 189 5.1 Introduction...................................189 5.2 PlanarPathGeneration ............................190 5.3 Planar Path Generation With Prescribed Timing . 192 5.4 CouplerCurvesofPlanarFour-BarLinkages . 197 5.5 TheTheoremofRoberts-Chebyshev. 201 A A Summary of Dual Algebra 203 A.1 Introduction...................................203 A.2 Definitions....................................204 A.3 Fundamentals of Rigid-Body Kinematics . 208 A.3.1 Finite Displacements . 209 A.3.2 Velocity Analysis . 217 A.3.3 The Linear Invariants of the Dual Rotation Matrix . 219 A.3.4 The Dual Euler-Rodrigues Parameters of a Rigid-Body Motion . 224 A.4 TheDualAngularVelocity. .229 A.5 Conclusions ...................................235 Bibliography 237 Index 247 iii iv Chapter 1 Introduction to Kinematic Synthesis 1.1 The Role of Kinematic Synthesis in Mechanical Design When designing a mechanical system, whether a structure or a machine, the first step is to produce a conceptual design that will meet the design specifications. Broadly speaking, the main function of a structure is to be capable of withstanding the anticipated loads without exhibiting major deformations that would hamper the integrity of the structure or the safety of its occupants. Likewise, the main function of a machine is to be able to perform the intended task, usually involving finite displacements of its parts, without major deformations that would hamper the integrity of the machine or the safety of its users. To be true, mechanical systems exhibit, more often that not, features of both structures and mechanisms. Such examples occur in transportation machinery, such as landing gears in aircraft. Figure 1.1 illustrates the structure of the deployed landing gear of the Airbus A300-600 aircraft. In this posture, the landing gear works as a structure, to withstand not only the static load of the airplane on the tarmac, but also the dynamic load exerted by the unavoidable impact upon landing. Transportation machinery is a domain in which mechanisms, especially linkages— the focus of this course—plays a major role. Shown in Fig. 1.2 is a depiction of the powertrain of a Class-C Mercedes Benz. A key subsystem of the system in question— the powertrain—is the steering linkage, whose main components are visible in the figure. Again, this mechanism plays the dual role of a structure and a machine, as its function is not only to properly orient the planes of the front wheels upon turning, but also to support the wheels and the loads transmitted by the ground onto the vehicle frame. In the above preamble we have introduced concepts of engineering design as pertaining to mechanical systems at large. Of these, we have focused on structures and machines. In fact, design, together with manufacturing, is the raison d’ˆetre of engineers, all disciplines 5 Figure 1.1: The landing gear of an Airbus A300-600 known as engineering science, namely, mechanics, thermofluids and numerics, to name but just the major branches, playing a supporting role in the production process. For this reason it is necessary to dwell on this concept. Because of its importance, the engineering design process has been the subject of study over the centuries, starting with Marcus Vitruvius Polio (ca. 75 BCE–ca. 15 CE) and his 10-volume work under the title De Architectura (Vitruvius, 28 B.C.E.). Modern engineering design theory owes its origins, to a great extent, to Franz Reuleaux (1829–1905), who first proposed a grammar to describe the kinematic chain of a machine (Moon, 2003). A modern model of the design process, due to French (1992), is depicted in Fig. 1.3. In this model, four stages are distinguished: a) analysis of the problem; b) conceptual design; c) embodiment design; and d) detailing, or detailed design. In the first stage, analysis of the problem, the functions required from the object under design, in our case, a machine, are clearly defined, in general, but precise terms. At this stage, the task of the design engineer is to produce a) design requirements, in terms as general as possible, in order to avoid biasing the design team towards a specific layout of the solution, and b) design specifications, so as to satisfy the rather vaguely spelled-out needs of the client. In the second stage, the design team produces a set of design variants, as rich as 6 Figure 1.2: A view of the powertrain of a C-Class Mercedes Benz possible, after several sessions, structured or unstructured, which are part of the creative aspect of the design process. In the third stage, the design team focuses on a reduced set of design variants, those having the highest likelihood of succeeding in satisfying the client’s demands within the resources—budget, milestones, technology—set at the disposal of the design team. In this stage, the task of the team is to produce a preliminary model of the design solution, with a clear identification of the main parameters defining a specific design variant. Further, a parametric model of each of the short-listed candidate variants is produced, which is then subject to optimization with the aim of finding the specific fundamental dimensions that either
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