Biomechanics (part 2)

Transcription

1 Biomechanics (part 2) MCE 493/593 & ECE 492/592 Prosthesis Design and Control September 11, 214 Antonie J. (Ton) van den Bogert Mechanical Engineering Cleveland State University 1

2 Today Coupling between muscles and skeleton Neural control

3 Joint Moments M( q) q C( q, q ) q g( q) J T F q,, F are vectors joint moments forces and moments applied from outside In the human body, joint moments are generated by 1. muscles 2. ligaments (mostly at end of range of motion) 3. prosthetic / orthotic devices Terminology: moment or torque?

4 Joint moment generated by muscle muscle a q b Muscle length L m is a function of joint angle q Cosine rule: L L 2 m m a 2 a b 2 2 b 2abcosq 2 2abcosq Mechanical power output of muscle: Power output of joint torque: FL m So, a joint torque is equivalent to a muscle force F if: dl m dq In this example: P P q is the moment arm or mechanical advantage of the muscle with respect to coordinate q dl dq m a 2 b ab sinq 2 2abcosq ab sinq L m FL m dl F dq dl dq q m q q m F

5 Anatomist s definition of moment arm distance between muscle and joint muscle a q b d ab sinq L m d engineer and anatomist agree on this one

6 In general (many DOFs and muscles) M T dlm ( q) q C( q, q ) q g( q) Fm other dq J T F ext q,, F m, L m, F ext are vectors dl m dq joint moments from muscles is a matrix of moment arms matrix element i,j is the moment arm of muscle i with respect to coordinate j forces and moments applied from outside (and you do it the same way with linear actuators in a prosthetic device!)

7 Muscle path does not have to be a straight line wrapping points wrapping surface If your model allows you to calculate muscle length as a function of q, and take the derivative, you have the moment arms. Opensim software

8 Muscle does not have to cross the joint to have a moment arm Fourbar linkage (for example: closed chain exercise) dl q m dq an anatomist would have some trouble with this idea is the moment arm in this example positive or negative?

9 Muscle can cross more than one joint And have a counterintuitive moment arm Andrews JG (1985) A general method for determining the functional role of a muscle. ASME J Biomech Eng 17: DOF: knee flexion angle q Hamstrings length is a function of q: L m (q) moment arm dl m dq is negative so hamstrings are a knee extensor! (when you have these constraints on foot and pelvis)

10 Moment arms can be measured Distance to joint (not the best idea) Move the joints, measure muscle length change L m (q), and take partial derivatives tendon travel method muscle length L multiple joints: partial derivatives joint angle [radians!!]

11 Two-joint muscles and energy transfer Gastrocnemius crosses knee and ankle Coordinates: knee flexion angle (q4), ankle plantarflexion angle (q5) During push-off phase of gait knee is extending ankle is plantarflexing Gastrocnemius produces knee flexion moment ankle plantarflexion moment q q Power: P P 4 5 (absorbs) (generates)

12 Analysis of normal gait This analysis does not consider muscles (joint torques only) Farris & Sawicki, J Royal Soc Interface 212 It shows that half of the required ankle power could have been transferred from the knee by the Gastrocnemius muscle. Quadriceps moves the ankle! May improve control! This transfer could be done by a fully passive elastic structure (but then you can t turn it off)

13 I used to think that Robotic systems needed elastic structures and actuators crossing more than one joint to match human control and efficiency But: elasticity and energy transfer can be achieved in the electric domain and controlled also

14 Closed loop system brain spinal cord sensory signals afferent nerves muscles muscle stimulation efferent nerves muscle forces skeleton sensory organs movement tissue mechanics

15 Neural activation neuron axon Action potential: a polarity reversal that travels along the axon Each neuron activates one motor unit with varying firing rate Action potentials travel along muscle fibers neuromuscular junction EMG (electromyography) is the resultant of all these action potentials as seen at the electrode

16 Time delays Multiple conversions between chemical and electrical signals especially in spinal circuit with multiple neurons polysynaptic reflexes Action potentials travel at 8-12 m/s From spinal cord to leg: 1 ms Muscle activation is a chemical process 2 ms to peak force, 4 ms to decay (depends on fiber type) muscle twitches at 15 Hz

17 Spinal circuits Central Pattern Generator (CPG) neural oscillator, sends rhythmic stimulation to muscles feedforward control Example: furnace with timer can be modulated by brain input exists in fish, insects, cats, infants existence in adult humans is controversial Sensory inputs needed for feedback control Example: furnace with thermostat -> equilibrium point hypothesis Neural transmission delay in lower extremity: 5-1 ms can entrain the CPG can assist learning of feedforward strategies reflex pathways have been well studied

18 Sensory organs Vision, hearing: not essential for low-level control of walking and standing Essential sensors: cutaneous (skin pressure) sensors muscle receptors (force, stretch) balance sensors Sensors send action potentials to the spinal cord afferent neurons

19 Muscle sensors Muscle spindles: respond to stretch & stretch velocity Golgi tendon organ: responds to force

20 Stretch reflex

21 Skin pressure receptors

22 Cutaneous (skin) reflexes Negative feedback: withdrawal reflex muscle action removes stimulus helps prevent stumbling Positive feedback muscle action increases stimulus cats stance control Sign of reflex gain can depend on phase of movement! Zehr & Stein, 1999

23 Balance linear and angular accelerometers

24 Vestibulo-ocular reflex

25 Reflex-based control No CPG, no clock! System is autonomous Geyer & Herr, IEEE Trans Neural Syst Rehabil Eng, 21 x x x f f f x, u x, u x, u t x open loop reflex based Herr invented the Rheo knee and the BIOM foot BIOM uses reflexes in some way

26 Simulation results You probably need realistic muscle mechanical properties to make this work The controller is a bit too hand-crafted for my taste optimization?

27 My work in 29 Open loop optimal control solution x O (t), u O (t) Feedback controller: u = u O (t) + G [ s s(x O (t)) ] Gain matrix (16 x 3) G = right side muscles left side muscles Gains feet muscle spindles angles ang.vel Signs fixed, positive ( ) or negative ( ) Same gain magnitude within each sensor type Model will follow trajectory x O (t) until perturbed

28 Formal stability analysis of limit cycle Dingwell & Kang, J Biomech Eng 27. Floquet analysis Quantify the growth/damping of perturbations from one gait cycle to the next Linearization: (x k+1 x*) = A (x k x*) Matrix A calculated from model Eigenvalues of A: Floquet multipliers λ (5) Floquet exponents: μ = log(λ)/t units: s -1 Movement is stable when Maximum Floquet Exponent (MFE) < δx(t)~e μt

29 Anecdotal stability analysis Perturb forward velocity by 2% Equivalent to impulsive force Simulate half a gait cycle By how much has the trunk fallen? Vertical Trunk Excursion (VTE) VTE initial state final state

30 TA TA Sol Sol [BW] Gas Gas Vas Vas RF RF Ham Ham [BW] Glu Glu Ilio Ilio [degrees] Open loop optimal control solution 3 2 Hip Angle 6 Knee Angle 1 Ankle Angle GRF Y Time [% of gait cycle] File name: Number of nodes: 1 Initial guess: Model used:./result1half.mat../7result.mat../../legs2dmex/ccfmodel Gait data tracked:../wintergaitdata.mat Weffort: 1 GRF X Norm of constraints: Cost function value: Muscle Forces Muscle Activations 5 1 Can be done with subject-specific model subject-specific gait data

31 Stability analysis of open loop controlled model Floquet multipliers λ Floquet exponents (s -1 ) μ = log(λ)/t

32 Floquet multiplier 1. Eigenvector = forward translation

33 Floquet multiplier (Floquet exponent 14.1 s -1 ) Eigenvector = generalized coordinates generalized velocities stance -.6 leg joint.2 angles muscle CE lengths muscle active states

34 Floquet multiplier 9.36 (Floquet exponent 3.5 s -1 ) Eigenvector = generalized coordinates generalized velocities muscle CE lengths muscle active states

35 Response of open loop controlled model to forward pull VTE = 16.6 cm final state

36 with feedback control added?

37 Max Floquet Exponent (s -1 ) Vertical Trunk Excursion (m) Muscle spindle feedback 2 Floquet.2 VTE Spindle gain (m -1 s) Spindle gain (m -1 s) gain = gain = 1.96 m -1 s

38 Max Floquet Exponent (s -1 ) Vertical Trunk Excursion (m) Joint angle feedback Floquet.15 VTE angle gain (rad -1 ) angle gain (rad -1 ) gain = gain =.7 rad -1

39 Max Floquet Exponent (s -1 ) Vertical Trunk Excursion (m) Joint angular velocity feedback 15 1 Floquet.2.15 VTE angular velocity gain (rad -1 s) angular velocity gain (rad -1 s) gain = gain =.22 rad -1 s

40 Max Floquet Exponent (s -1 ) Vertical Trunk Excursion (m) Forefoot pressure feedback Floquet.2.15 VTE GRF gain (N -1 ) x GRF gain (N -1 ) x 1-3 gain = gain =.138 N -1

41 Effect of simple feedback Feedback from each type of sensor could improve stability Agreement between Floquet analysis and finite perturbation response An optimal feedback gain always existed Stability (MFE<) was not yet achieved Feedback from combination of sensor types?

42 Max. Floquet Exponent (s -1 ) Combined feedback 1 5 MFE (s -1 ) angle gain (rad -1 ) angular velocity gain (rad -1 s) Lowest MFE:.1482 s -1 Angle gain 1.4 rad -1 Angular velocity gain.12 rad -1 s

43 Continuous walking with optimal combined feedback? Why not stable, as predicted by MFE? Limitations of Floquet analysis Accuracy Linearization around optimal trajectory (small foot clearance!)

44 Limitations of control system Sensors All sensors in one group had same gain Only some sensor combinations were tested Missing sensors Vestibular, etc. Physiological feedback is not always linear Threshold effects Reflex modulation Stumble response

45 Human stability tests Able bodied subject Impulsive force 1% BW for 2 ms (Δv.2 m/s)

46 2 1 Human stability test Gait analysis data (Trial 25,.85 m/s) R GRF L GRF GRF.4.2 R GAS EMG R.Gastroc L GAS EMG L.Gastroc R VL EMG R. Vastus Lateralis R AnkleMoment 2 L AnkleMoment L VL EMG L. Vastus Lateralis Ankle moments