PowerPoint Presentation• Goal: improve human movement
movement” data
• Prototyping and human testing needed
• Predictive simulation of human movement
can speed up the development process
• Results must be good enough
• Results must be obtained quickly
Simulation of movement
• Simulation is normally done forward in time, e.g.
• This is appropriate for many applications
• But not always…
variable step solver
• movement is optimal (e.g. minimal energy)
• and satisfies task constraints (e.g. initial and/or final state)
• State of the art: Anderson & Pandy (ASME J Biomech Eng 2001)
• 3D musculoskeletal model
• objective: metabolic energy
• 810 unknowns (54 muscles,15 time points)
• repeated simulations (“shooting”)
• final state x(T) still had an error
Spacetime methods
physics-based animation
“Because we extend the model through time as well as space, we
call the formulation spacetime constraints.”
The main idea
• Iterate x(t) and u(t) until:
• movement is optimal
• satisfies task constraints
• satisfies physics constraints:
u)f(x,x
• System dynamics:
• Initial state
• Final state
x uxmgdxI sin
• Cost function becomes an algebraic function:
• Time step h usually much larger than in ODE solver • Convergence study to decide how many nodes are needed
• For human gait cycle: N=50 or N=100 is typically good enough
uxmgdxI sin ii iii uxmgd
h
• fmincon sequential quadratic programming
TNN uuuxxx 2121 ,,,y
Matlab demo
GAIT
• 16 Hill-based muscles
• Matlab MEX function
• Find optimal gait
• direct collocation on N time nodes per half gait cycle:
• 66N unknowns, collected in a large vector
• typically N=50 (determined by mesh refinement) → 3300 unknowns
Ackermann & van den Bogert, J Biomech 2010







Tyy )( 33001y
Method details: constraints
• 50 task constraints, half gait cycle T and walking speed V
• 50(N-1) constraints due to system dynamics
• Total: 50N constraints c(y) = 0
Ackermann & van den Bogert, J Biomech 2010
x
u)f(x,x 2
• General form:

exponent p = 2,3,…
Ackermann & van den Bogert, J Biomech 2010
cost function weighted by
Effect of gravity
EARTH MARS MOON
• as long as tracking term is small














Combined human/device model
• Hydraulic energy-storing knee
prosthetic side (blue curves):
but it is the ankle’s fault!
Implicit formulation
0, u)xf(x,
Computation time ~ 7 min (2 min with good initial guess)
19 runs with random initial guess
• 3 runs did not converge
• 3 runs found the same solution (with cost 0.07)
• was also found from “intelligent” initial guesses
• 13 locally optimal solutions with higher cost
cost 0.071 cost 0.158
• Walk with a somersault in each step:
• Solution could only be found from random initial guess
31 ˆ2ˆ0 xxxx TvT
done without tracking human motion data
• direct collocation method was presented
• can include “hard” task constraints
• Many unknowns: discretized state and control trajectories • must use gradient-based solvers
• can fail to converge
• can even fail to produce a movement that is physically possible
• even if converged, solution can be a local optimum