Physics-based Simulation in Sports and Character Animation Kuangyou Bruce Cheng ( 鄭匡佑 ) Institute of Physical Education, Health, & Leisure Studies National Cheng Kung University, Tainan, Taiwan

Outline: Introduction to simulation Previous and current research topics Summary of methods and results Discussion and Conclusions Additional topics

What is a simulation? Reproduce a real event with a different (usually simplified) approach For example, free-fall bodies; multi-segment rigid body systems Generation of equations of motion (based on physics laws) Numerical solution of ODE/PDE Advantages: lower cost, no risks, repeatable and no human-related error factors

Previous research topics Jumping from a compliant surface (application to springboard diving jumps) Standing vertical jump (effect of joint strengthening and effect of arm motion) Standing long jump (effect of different starting posture and additional weight) Optimal flight trajectories of the shot-put and discus

Current research topics Multi-stage simulation and optimization of jumping (swim start, standing long jump, ski jump, vaulting) Biomechanical analysis of Tai Chi Push- hand Muscle/joint onset sequence in fast reaching movements Physics-based movement simulation of animated characters

Forward and Inverse Dynamics F = ma From the driving forces/torques, what are the resulting motions? From the observed motions, what are the driving forces/torques?

Motivations for doing forward simulation The best control strategies for many human movements are not clear Real subjects’ performance may be affected by practice and psychological factors Computer simulation with optimization serves as a promising tool Very few studies considered multi-stage simulation and optimization

Summary of previous researches Discus Flight

Optimization problem formulation: Goal: Maximize flight distance by optimizing the initial release angle and two orientation angles (with fixed release speed and height). Objective function: Flight distance can be calculated by numerically solving ODE’s with known equations of motion and initial conditions.

Simple model of springboard jumping Massless leg (length = 2a) Straight leg at θ = 180 deg mdmd m b g k T x1x1 x2x2 θ a a

Instantaneous joint torque T(t) depends on maximum isometric torque T max and 3 variables: f(θ) θ Angle dependence Angular velocity dependence (according to Hill’s muscle model)

Resultant effect of related muscle activation Inputs to actuate the model Node points representation A(t) ranges from -1 (full-effort flexion) to +1 (full-effort extension) Time constant approximated from rise and decay time constants for muscle activation Knee torque activation level: A(t)

Multi-segment 2-D models (Equations of motion derived by Autolev) Trunk & head Arms Thigh Shank Feet Springboard Torque generators at ankle, knee, hip, and shoulder (5-segment)

Optimization problem formulation: Goal: Maximize jump height by optimizing joint activation nodal values during contact. Objective function: J = y + v 2 /2g, where y and v are COM vertical position and velocity at takeoff.

Optimization Implementation: Parameter optimization: node points are to be optimized (since they represent joint torque activation level) Algorithm: Downhill Simplex method (with different initial guesses for more reliable optimal solution)

Results overview: General agreement between optimal simulated and measured motions Coordination strategies (joint torque activation patterns) different from those in rigid-surface jumping Predicted optimal fulcrum setting (board stiffness) is in agreement with experiment

Results overview (continued): Kinematic and coordination characteristics in jumps maximizing somersault rotations differ from those in pure jumping Arm motion has significant effect on generating more angular momentum

Results of discus flight: Right: Optimal initial conditions Left: Effect of wind

Results from simple model: Simulated optimal jumps with constraint θ ≥ 90 deg (S90), and measured jumps; board tip (―) and diver c.m. (x) position vs. time

Optimal simulated springboard jumping (4-segment model) Stick figure animation plotted using MATLAB

Comparison of simulated and measured jumps

Simulated joint torque (―) and joint activation level (x). Joint torque is normalized by dividing its value by maximum isometric torque.

Jump height vs. fulcrum setting

Jumping for maximizing backward somersault rotations (4-segment model):

Jumping for maximizing backward somersault rotations with arm swing:

Combining with the flight phase

Results of multi-stage simulation and optimization for ski jumping

Summary of some current researches Modeling and experimental validation of swim diving

Walking Animation with Inverted Pendulum Model g single support Center of Mass(COM) g Pivot Pivot Massless pendulum Mass

Inverted Pendulum Model g Pivot Pivot Massless pendulum Mass

Inverted Pendulum Model C onservation of Energy g θsθs θeθe r where single support g θsθs θeθe

Inverted Pendulum Model Velocity changes at the double support phase g V0V0 V1V1 V2V2 α α single support double support

Discussion: Adequacy of studying human movement from simulation and optimization approach Maximal joint torque activations are timed to occur around maximal board deflection when the board is best able to resist An optimal surface compliance for jumping exists (not a psychological effect)

Discussion (continued): When arm motion is restricted, optimal jumping for backward somersaults involves partially extended knee and fully extended hip With arm motion, however, the knee is fully extended to create larger angular momentum at takeoff Multi-stage simulation/optimization is necessary since movements in the current stage affect those in the following stage Real-time responsive human walking can be simulated with a simple inverted pendulum model

Conclusions: Significance of simulation and optimization approach for studying human movements Joint torque activation strategies are different for different movements and should be subject- specific Advantages in investigating movement sensitivities to parameter changes (that cannot be tested in real subjects) Other movement application

Additional topics Musculoskeletal dynamics Gluteus maximus Hamstrings Gastrocnemius Rectus femoris Vastus group Soleus Other plantarflexors Tibialis anterior.

Musculotendon model

Musculotendon activation dynamics L: musculotendon length V: musculotendon velocity a: muscle activation F: musculotendon force

Neural excitation dynamics u(t): net neural control signal (0 < u(t) < 1)  rise (22 ms) and  fall (200 ms) are the rise and decay time constants for muscle activation.

Forward simulation with muscles : Combine excitation, activation dynamics, and knowledge of muscle insertion locations Forward simulation with node points of u(t) (muscle excitation) and final time as inputs Find node points that optimize the performance criterion

Pedaling animation (Neptune and Hull, 1999)

Thank you for your participation Questions