Dynamics of Multibody Systems: Conventional and Graph-Theoretic Approaches

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1 Dynamics of Multibody Systems: Conventional and Graph-Theoretic Approaches SD 65 John McPhee Systems Design Engineering University of Waterloo, Canada

2 Summary of Course: 1. Review of kinematics and dynamics. Conventional multibody dynamics 1. Planar systems. Spatial systems and MSC.Adams 3. Graph-theoretic modelling: 1. One-dimensional systems. Multibody systems and MapleSim 4. Advanced topics

Spatial systems and MSC.Adams 3. Graph-theoretic modelling: 1.

3 1. Multibody Mechanical Systems a collection of rigid and flexible bodies connected by joints, e.g. Serial Robots Parallel Robots (PKMs)

4 Walking robots:

5 Mechanisms and machinery: Spatial slider-crank mechanism

6 Vehicles (road, rail, aerospace): Lola World Sports Car

7 Multibody system dynamics: given only a description of the system as input, formulate the kinematic and dynamic equations needed to determine the system response. System Equations Dynamic simulation Sensitivity + optimization Model-based control

equations needed to determine the system response.

8 Example multibody system: Planar slider-crank mechanism (f=1dof)

9 Coordinates: Must be defined a priori Selection affects the number and nature of equations Absolute coordinates: Position and orientation of every body in system, e.g. q = [ θ3 x, y, θ, x, y, θ, x, y, ] Easy to formulate equations automatically Very large systems of equations Joint (relative) coordinates: Correspond to joints in system, e.g. T q = [ θ, β, s] Fewer in number (minimum for open-loop systems) Requires more topological accounting T

q = [ 1 1 1 3 3 θ3 x, y, θ, x, y, θ, x, y, ] Easy to formulate equations automatically Very large systems of

10 Kinematic Analysis: One prescribed motion per dof, e.g. if θ = f(t) for slider-crank, solve the kinematic constraint equations for q(t)=[θ,β,s] T : L1 cosθ + L sin β s Φ ( q, t) = L1 sinθ L cos β = θ f ( t) 0 For velocities, solve: Φ q q = Φ t Φ q = L1 sinθ L1 cosθ 1 L L cos β sin β

for q(t)=[θ,β,s] T : L1 cosθ + L sin β s Φ ( q, t) = L1 sinθ L cos β = θ

11 Dynamic equations from Newton-Euler, or from the Principle of Virtual Work: δw = T + T T δθ F δr + δw + RB δw n T δw δw RB FB = ma = δr n T F Expressing all variables in terms of q: n RB n FB T δr ( Iω + ω Iω) δθ T ( f ρa) dv δε σ dv T b V V FB = 0 T δw = Q δq = 0 Set the n generalized forces Q=0: Mq + Φ T q λ = F

variables in terms of q: n RB n FB T δr ( Iω + ω Iω) δθ T ( f ρa) dv δε σ dv T

12 Where, for the planar slider-crank: M = m1l1 0 0 / 3 m L 0 m L / 3 cos β / m L m 0 cos β / + m 3 F = [ ] 0, 0, F m L β sin / T β Φ q = L1 sinθ L1 cosθ L L cos β sin β 1 0 A dynamic simulation is obtained by solving the n+m differential-algebraic equations for q(t) and λ(t).

L1 sinθ L1 cosθ L L cos β sin β 1 0 A dynamic simulation is obtained

13 . Conventional Methods for Multibody Systems Automated modelling and simulation Based on absolute coordinates Large systems of nonlinear DAEs Numerical data, not symbolic equations Commercial software: Working Model MSC.Adams

Large systems of nonlinear DAEs Numerical data, not

14 4 th -year (senior) design projects: Hexplorer 6-legged walking robot

15 Hexplorer 3-DOF leg in extended position

16 ADAMS simulation of walking maneuver

17 SAE mini-baja vehicle: Winner of ADAMS modelling award

18 Research into vehicle stability: Grapple Skidder (Timberjack Inc)

19 Research into vehicle stability: ADAMS simulation of roll-over

20 Research into mechanisms and machinery: 6-bar mechanism designs

21 Research into vehicle suspension design: Lola World Sports Car (Multimatic Inc)

22 ADAMS model of four-post test:

23 no chassis flexibility or joint compliance rear left-hand suspension:

24 Research into biomechanics: Investigation of metabolic energy consumption for normal and prosthetic gaits.

25 Research into biomechanics: Forward dynamic simulation, realistic friction

26 Research into biomechanics: Forward dynamic simulation, low friction

27 3. Modelling using Linear Graph Theory Origins in Koningsburg, Prussia, 173:

28 Topology = 4 land masses connected by 7 bridges: Leonhard Euler s sketch of Koningsburg topology

29 Euler s linear graph representation of topology: C c g A d e D a b f B If there are more than nodes with odd valence, then an Eulerian path cannot exist, Euler (173)

30 Graph-Theoretic Modelling (GTM) Component models from measurements α,τ Edge = element, Nodes = connection points Through variable τ = current ( i) Across variable α = voltage ( v) Constitutive equations, e.g. v = L di dt

31 System model from e assembled components: Electrical Circuit and Linear Graph e constitutive equations (linear or nonlinear) e linear topological equations from system graph primary variables determined by tree selection

32 Linear graph (nodes=frames, edges=elements): τ = ( F, T ), α = ( r, θ ) Vector variables: Cutsets = dynamic equilibrium for a subsystem Circuits = summation of vector displacements around closed kinematic chains

33 Advantages of G-T modelling approach: Very systematic Amenable to computer implementation Leads to efficient systems of equations Suitable for real-time simulation, e.g. for virtual reality or hardware-in-loop experiments Applicable to multiple physical domains Unifying: coordinates may be absolute or joint or other possibilities

34 n branch coordinates q are defined by tree selection: T T y x y x y x m m m r r Tree s s h h r r Tree ],,,,,,,, [ },,, { ],, [ },,, { θ θ θ β θ = = = = q q

35 Mechatronic Multibody Systems

36 Robot control system to capture moving payload tracked by overhead vision system:

37 Electrical network + multibody system, coupled by transducer elements:

38 Linear graph representation: The DC-motors (transducers) have an edge in both the mechanical and electrical sub-graphs.

39 the equations for the individual domains are obtained using the electrical and multibody formulations. these equations are coupled by the constitutive equations for the transducers, e.g. for the DC-motor: v = K T = K v T dθ + Ri + dt dθ i B dt from a single linear graph representation, the governing equations are automatically derived in symbolic form by a Maple program DynaFlexPro, now part of the MapleSim package. L di dt

40 Maple Algorithms (MapleSim) Flowchart: 1) Build Model Model Description (ASCII File - *.dfp) ) Build Equations Mathematical Model (Maple Module) 3) Build Sim Code Optimized Simulation Code

41 Human and Robotic Slapshots Motion capture analysis of slapshot

42 Human and Robotic Slapshots Downswing Puck contact Synthesis of 4-bar hockey robot Thor

43 Human and Robotic Slapshots MapleSim model

44 Human and Robotic Slapshots MapleSim animation

45 4. Advanced Topics: Modelling of mechatronic systems Modelling of contact dynamics Vehicle dynamics and tires

46 Mechatronic system models: Condensator microphone (Hadwich and Pfeiffer, 1995)

47 Linear graph representation: where, for the moving-plate capacitor, i F = C = 1 dv ( s) + dt dc( s) v ds dc( s) ds ds dt v

48 Selecting the trees shown, and using the current formulation, one obtains equations in terms of i ( ) and s(t) t C 5 ( s) m s + dv dt d 6 s + + dc( s) ds k 6 s + m g 5 ds dt v 1 i = 0 dc( s) ds v = 0 where: di v + dt ( = R1i L3 E4 t )

49 Mechatronic system models:

50 Linear graph of multibody system + induction motor :

51 Rotation of input link:

52 Current through one rotor inductor:

53 Contact dynamic models:

54 Discrete versus continuous models [Gilardi and Sharf] Hunt-Crossley model with modified damping: f ( + a ) p n = k xn 1 xn Sphere dropped inside cylinder:

55 Volumetric contact model [Gonthier et al, 006]: f k ( + a ) f n = V 1 xn hf Rolling resistance, etc, also a function of geometry:

56 Volumetric contact model [Gonthier et al, 006]:

57 Contact dynamic applications:

58 Vehicle dynamics and tires:

59 Application to planetary rovers [Petersen, 011]: Juno rover MapleSim model

60 Application to planetary rovers [Petersen, 011]: MapleSim model, with tire/soil interactions

61 Application to planetary rovers [Petersen, 011]:

62 Application to planetary rovers [Petersen, 011]:

63 Dynamics of Multibody Systems: Conventional and Graph-Theoretic Approaches SD 65 John McPhee Systems Design Engineering University of Waterloo, Canada

2.1 Introduction. 2.2 Terms and definitions

2.1 Introduction. 2.2 Terms and definitions .1 Introduction An important step in the procedure for solving any circuit problem consists first in selecting a number of independent branch currents as (known as loop currents or mesh currents) variables,