Multibody dynamics and numerical modelling of muscles LORENZO GRASSI

Transcription

1 Multibody dynamics and numerical modelling of muscles LORENZO GRASSI

2 Agenda 10:15 Lecture: Introduction to modelling locomotion and inverse dynamics 11:00 Break/questions 11:15 Opportunities in the biomechanics group 11:30 Handing back assignment 1

3 Multibody dynamics and numerical modelling of muscles LORENZO GRASSI

4 What is multibody dynamics? A multibody dynamic (MBD) system is a system that consists of solid bodies, or links, that are connected to each other by joints that restrict their relative motion.

5 Why multibody dynamics?

6 Why multibody dynamics?

7 Why multibody dynamics? 1) 2) Skeleton of a baseball pitcher during the different phases of a pitch (3) 3D musculotendinous model to simulate the biomechanical effects of rectus femoris transposition 3) 1. Chao, E.Y. Med Eng Phys, (3) 2. Asakawa, D.S., et al. J Bone Joint Surg Am, A(2) 3. Leardini, A. et. al. Gait & Posture 26 (2007) Kinematic analysis for the rehabilitation planning

8 Schematic of a multibody system The human body is modelled as a number of rigid bodies connected by ideal joints......remember assignments 1 & 2?

9 Your (very) first taste of multibody dynamics Assignment 1 and 2

10 Different types of joints in our body

11 Assignment 1: from motion to forces In our assignment 1, the human leg was modelled with: - 2 rigid bodies (upper and lower leg) - 2 hinges 2 dof - movements limited to the sagittal plane... Kinematics data were used to calculate joint forces (but muscles were not considered)

12 3D is way more complicated! l 6( n 1) n (6 k 1 lk ) (Gruber) Ball and socket (3 dof) l = number of degrees of freedom of a system n = # rigid bodies Hinge (1 dof) l k = degrees of freedom for the k th joint For our 3D model: Hinge (1 dof) l 6(7 1) 6 k 1 (6 36 2*3 5*4 10 lk )

13 Assignment 2: redundancy & recruiting Several muscles act on the same dof: 1. Performing the same joint motion (synergist) 2. Neutralizing each other (antagonists) Features: 1. Repeated movements produce similar activation patterns pre-defined control strategies exist? 2. When the articular load increases, so does the muscular activation, up to the tetanic limit External forces were known, and we used a static optimization approach to calculate muscular forces

14 Why are we redundant? 1. Increase articular stability Weight lifting, execution of new motor tasks, instability. 2. Transferring forces/moments between joints Co-contractions at the hip can produce an increment of the bending moment of the knee (e.g., co-contraction of gluteus maximus and rectus femoris produces knee extension). 3. High accuracy movements Highly accurate and precise finger movements require complex activation patterns 4. Improve movements that require changes in direction 5. Protect the joints in extreme articular positions

15 M f 0 Static optimization i int f F f F i m R( q) F i m ( F 1 m, F F max 0 2 m,..., MT F n m ) M int = moment equilibrium equations f = cost function f i F i m PCSA i n n = 1 not effective in predicting synergies, especially for low load magnitudes n > 1 synergies are predicted, but additional constraints are necessary to avoid muscular overloads n synergies are maximized, and effort is minimized All the exponents n > 1 predict synergies, but fail at predicting antagonisms

16 Assignments 1 & 2 were just the first taste of the multibody dynamics now let s go deeper into the topic

17 Generation of the body motion 1. Excitation 2. Activation 3. Force 4. Joint torques 5. Dynamics of the rigid body system Gravitational effects BODY MOTION Joint moments due to external forced (e.g. ground reaction) M ( q) q C( q) q 2 G( q) R( q) F MT Me( q, q ) 0 Mass matrix Joint moments due to muscle forces Centrifugal and Coriolis effects

18 Different approaches are possible Assignment 1

19 Forward dynamics ), ( ) ( ) ( ) ( ) ( 2 1 q q Me F q R q G q q C q M q MT Looks like a very nasty equation to solve and it is! But computers can help us with its solution!

20 Numerical modeling of the tendon and muscle mechanics

21 The modeling approach There are two possible numerical descriptions: a phenomenological one (Hill, ), and a mechanicistic one, based on physiology and the biochemistry of muscular contraction (Huxley, ) Simple mathematical expressions, based on measurable parameters Differential equations, with several parameters hard to quantify

22 Muscle model (Thelen, 2003) CE = contractile element α M = pennation angle The muscle force generated is a function of three factors: the activation value (a), the normalized length of the muscle unit, and the normalized velocity of the muscle unit.

23 Muscle model (Thelen, 2003) CE = contractile element α M = pennation angle

24 Muscle model (Thelen, 2003) a) The relation between active force versus length can be described as a Gaussian. The relation between passive force and length has a first exponential phase, followed by a second linear phase b) scarico. b) The relation between active force and velocity can be scaled in order to reduce the contraction velocity in sub-tetanic conditions c) The force-strain relation for the tendon has a first exponential phase, followed by a linear phase

25 Muscle activation dynamics da dt a u a ( a, u) Where τ a (a,u) is a time factor which varies with the activation level, a is the muscle activation, and u is the excitation signal (Thelen, 2003). A more refined model could include different τ for rise and fall da dt rise u ua u a fall ;

26 Active muscle force f e l M ( L 1) 2 Where: f l is a scale factor L M is the normalized muscle length γ is a shape factor

27 Passive muscle force Where: F T F k toe e k T toe lin k toe ( e 1 T ( T T toe T toe ) F T toe T 1); T toe T ; T toe T toe F 1. is the tendon force normalized by the max isometric force 2. is the tendon deformation 3. is the limit elongation over which it behaves linearly 4. k toe is a shape factor 5. k lin is a scale factor. T 6. F toe 0.33 is the limit normalized force over which the tendon behaves linearly T T T toe

28 Muscle force Vs. velocity V M ( a) V M max F M b af l M V max is the max contraction velocity, and b is calculated differently whether the muscle fiber is shortening (F M <af l ) or lengthening (F M > af l ) b M af F l A M (2 2 / Af )( af l Flen M ( Flen 1) f ; F F M M af ) ; F l M af l M F len is the max normalized muscle force when the fiber is elongated af is a shape factor

29 So, let s get back to our forward dynamics problem

30 Forward dynamics Now we have all the theoretical background to start playing with our multibody dynamics software

31 Open-source software for the multibody simulation of the neuromuscular system and the motion dynamics simulation (numerical methods for the coupled solution of the multibody dynamic problem and the optimal distribution of the muscle forces) Website: There you can download and install the software for, and find a lot of tutorials and instructions SimTK and SimBios are trademarks of Stanford University

32 The GUI

33 Our test case: simulation and prevention of ankle sprain

34 Research questions of our test case You will examine and address how the following factors may affect angle inversion sprain injury: Muscle reflexes Muscle co-activation Introduction of a passive orthosis

35 When & Where Monday, September 19 th (group 1) 20 th (group 2), 10:00-12:00 Room INA 1-2 in the M:house Please be there on time!

36 Time for a break!

37 Reklam

38 Biomechanics group: who are we?

39 Biomechanics group: what can we give 1 more course: Tissue Biomechanics (BMEN10, HT2) Lecturer: Hanna Isaksson/Sophie Le Cann Mechanobiology of skeletal tissues (bone, cartilage, tendons and ligaments) More research oriented Combined experimental & numerical (FE) lab validation Many Master thesis projects Each of our research areas have available Master thesis projects FE modelling, experimental Soft & hard tissues You can find some examples of completed M.Sc. projects on our website ( We are also open to your suggestions: what would you like to do?

40 Handing back assignments

41 Handing back assignments Assignments marked by Lorenzo: Pencil LG signature means wrong (Swedish style)* Assignments marked by Neashan: Red pen No signature means correct (Rest of the World style) *

42 Handing back assignments For those whose assignment needs supplement: No worries, you have one more attempt If the corrections are not clear, ask me or Neashan ( ) Assignment 1 gives 1 point (out of 60) You have to complete the assignment anyway (even if the 2 nd submission is not correct, yet) Some suggestions to do better the next time: Consistency: stick to the chosen reference system Common sense: check if the obtained values make sense Compare the obtained forces with the subject s body weight

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