Reasoning about Human Motion: A Tutorial Overview Andreas Hofmann

Contents Introduction Architectures for motion control Impedance control algorithms Explaining and reproducing human motion trajectories Disturbances - classification and strategies Conclusion

Different kinds of human motion

Artificial human motion: Robotics

Artificial human motion: Animation

Artificial human motion: Assistive Devices

Goals of Motion Research Understand biomechanical principles underlying control of motion –Biomimetic motion trajectories –Stability in presence of disturbances Develop models and controllers based on such principles Develop estimators based on these models –Compute trajectories, intent, from noisy sensor input (video, EMG, marker)

Contents Introduction Architectures for motion control Impedance control algorithms Explaining and reproducing human motion trajectories Disturbances - classification and strategies Conclusion

Motor control – functional hierarchy High-level goals (conscious, flexible) –Win point vs. conserve energy (in tennis) Strategic planning (conscious, flexible) –Note opponent’s position, return to right rear baseline Tactical objectives (preconscious, learned) –Contact of racket with ball Action (subconscious, automatic) –Desired force and position trajectories Muscle activation

Motion execution architecture

Motion estimation architecture

Contents Introduction Architectures for motion control Impedance control algorithms Explaining and reproducing human motion trajectories Disturbances - classification and strategies Conclusion

Control of walking machines

Virtual Model Control Impedance control of reaction point –Reaction point is control point of interest (COM, swing leg, end-effector) –Virtual elements used to express setpoint, impedance of reaction point Compute desired force for reaction point –Jacobian used to compute joint torques that achieve desired reaction point force

Virtual Elements Specify translational and rotational force between action frame and reaction frame Force usually computed based on simple PD control law

Using Jacobian to Compute Joint Torques Simple Jacobian-based equation Derived using notion of virtual work Jacobian computed from kinematic transforms

Model Kinematics Sagittal Plane Hp, kp, ap are hip, knee, and ankle pitch Frontal Plane Hr, ar are hip and ankle roll Horizontal Plane Hy is hip yaw Joint angle vector for one leg:

VMC Jacobian Transformation – Single Leg Jacobian used to relate reaction frame force to joint torques - Based on virtual work derivation (Paul, Craig) so (simple equation requiring only matrix multiplication)

Caveats for torque computation using Jacobian -No limit on joint torques -(to model actuator limits, or to prevent foot slip or roll) -Not guaranteed to work for all poses -If leg completely straight, hp, kp, ap are all 0 -Jacobian not of full rank -Vertical force not feasible in this pose -Joint torques for hp, kp, and ap will be computed as 0, even if the desired vertical force is non-zero -In practice, such problems are avoided by bending the leg slightly in the computation

Hardware implementation Requires use of special actuators –Series-elastic actuators

Jacobian Computation for Single Leg Forward kinematic transform built up using chain of homogeneous transforms for segments and joints Ankle pitch transformShin transform Elements of J computed using partial derivative method (Paul, Craig)

Biological inspiration: equilibrium point hypothesis Motion as sequence of poses (Bizzi, et. al.)

How can virtual element parameters be determined? ”Intuitive” approaches are not adequate for locomotion applications

Contents Introduction Architectures for motion control Impedance control algorithms Explaining and reproducing human motion trajectories Disturbances - classification and strategies Conclusion

Achieving biomimetic motion What should the parameters for the virtual elements be? –“Intuitive” control (tweaking parameters by hand) is tricky Investigation of human motion and its underlying biomechanical control principles can provide useful guidance

Planar arm motion Flash and Hogan, minimum jerk Hasan, minimum effort Both give good predictions (within noise range of biological data)

3-D arm motion Atkeson and Hollerbach, invariants of 3-D arm motion Trajectories are curved, but –Invariant tangential velocity profile when normalized for speed and distance –Close to minimum-jerk Hypothesized association with visual kinematic coordinate frame

Challenges with locomotion Kinematic principles alone generally not adequate Foot contact as semi-underactuated joint Inverted pendulum analogy Dynamics are important

Dynamic optimization techniques Dynamic programming –Very general, can incorporate discrete as well as continuous variables, limitations with large state spaces, discretization Space-time Dynamic Optimization –Witkin, Popovic brothers –SQP, control points

Space-Time Dynamic Optimization Zoran Popovic animations

Space-Time Dynamic Optimization

Angle position trajectories represented as splines with control points –Typically, 10 – 20 control points for a 1 – 5 second trajectory Easy to differentiate to get velocity and acceleration splines Control points are parameters to be optimized –SQP algorithm (constrained non-linear optimization) –Kinematic, dynamic constraints –Cost function in terms of state and force variables

Space-Time Optimization Kinematic constraints –Relations of joint angles, angular velocities, and angular accelerations to segment positions, velocities, and accelerations –Minimum and maximum values for angles –Keep body upright –Keep swing foot off ground

Space-Time Optimization Dynamic constraints –Relation of joint torques to angular accelerations –Associated ground reaction forces, center of pressure

Important Biomechanical Principle Noticed interesting characteristics in test data collected at Spaulding Rehab Gait Lab Hypothesize that conservation of angular momentum about COM is being actively and vigorously asserted (closely controlled) for many kinds of movements

Assuming strict conservation of angular momentum, torque about COM must be 0. Therefore, by torque balance,

Conservation of Angular Momentum about COM This suggests a technique for computing COP if a desired COM trajectory is known Differentiate COM trajectory twice to get desired stance leg(s) ground reaction forces Use previous equation to compute corresponding COP

Conservation of Angular Momentum about COM Tested this hypothesis using human test subject Tests performed as Spaulding Rehab Gait Lab Developed full-body kinematic model corresponding to dimensions of test subject –Used to compute COM [Show test subject model movie] Following plots show results averaged over 5 trials –(1/2 gait cycle, right toe-off to left toe-off

Conservation of Angular Momentum about COM Tests show that COP can be predicted based on COM trajectory Foot placement can be predicted based on COM trajectory Suggests that dynamic optimizer should track COM, and derived ground reaction force and COP trajectories

Reduced-order Models

Sagittal Plane Model Dynamic Optimization Initialize all trajectories to straight-line interpolation from initial pose to final pose angle values Run dynamic optimization to get open-loop prediction of trajectories Compare with biological results [Play 100 movie.] Following plots show results for toe-off to heel- strike

Dynamic Optimization Results Discussion Starting from desired COM trajectory only - COM, COP, ground reaction forces from model all match biological results Stance leg angles match biological results Swing leg angles are not quite close enough –Swing foot position trajectory doesn’t match biological one closely enough

Dynamic Optimization Results Discussion Possible problems –Not tracking desired trajectories closely enough (Not enforcing constraints enough) –Missing a constraint, or some element of cost function Know, from validation, that model should be good enough with right constraints and cost function Separate issue – add toe joint to model

Dynamic Optimization Results Discussion Starting from desired COM trajectory only - COM, COP, ground reaction forces from model all match biological results Stance leg angles match biological results Swing leg angles are not quite close enough –Swing foot position trajectory doesn’t match biological one closely enough

Dynamic Optimization Results Discussion Possible problems –Not tracking desired trajectories closely enough (Not enforcing constraints enough) –Missing a constraint, or some element of cost function Know, from validation, that model should be good enough with right constraints and cost function Separate issue – add toe joint to model

Contents Introduction Architectures for motion control Impedance control algorithms Explaining and reproducing human motion trajectories Disturbances - classification and strategies Conclusion

Disturbance Reaction Types of balance disturbances –slipping (un-anticipated translation of foot in x-y direction) –rolling (un-anticipated rotation of foot, twisting ankle, for example) –tripping (over small obstacle, rock, for example) - un- anticipated impedance to trajectory of swing leg –being pushed (un-anticipated external force acting on some part of body, or on legs)

Disturbance Reaction to Slipping BALDER Platform –Boston University, Lars Oddssen, NeuroMuscular Research Center, Injury Analysis and Prevention Lab –Platform can exert 2G’s of force over a 1 foot displacement in the x-y plane –Simulates slipping (on a patch of ice, for example) –Platform includes walkway to allow for startup and recovery over an extended number of walking cycles –Platform has force plate. Marker data for head, torso, thighs, and shanks –Tested with healthy subjects, and patients with vestibular ailments

Disturbance Reaction to Tripping Treadmill with obstacles –University of Waterloo, Aftab Patla –Department of Kinesiology –Gait and posture laboratory –Reactive and proactive control of human locomotion

Disturbance Reaction to Foot Roll and to Push It may be difficult to obtain good demonstration data for these sorts of disturbances

Disturbance Reaction Based on Magnitude of Disturbance Small disturbance can be handled by ground force in double support, or even single support – use VMC controller with same biological virtual elements as those used for ordinary walking Larger disturbance requires dynamic, inertial movements –Of upper body, perhaps arms and swing leg –Additional virtual elements needed –Integration of position control for “swinging” limbs with force control of stance limbs Even larger disturbance requires shuffling of feet –Foot placement, step generation –Finite state machine deviates significantly from ordinary walking cycle

Disturbance Reaction Based on Magnitude of Disturbance Small disturbances –Finites state machine behaves no differently than for ordinary walking cycle (same states, same biological virtual elements, same parameters). –Existing biological elements (egg and wheel, for example) act in reflexive way to compensate for disturbance Medium disturbances –Finite state machine introduces additional biological virtual elements to control swinging of arms, swing leg, torso Large disturbances –Finite state machine introduces additional states in gait cycle to generate compensating steps (shuffling, etc.)

Disturbance Reaction – Distinction Between Reflexive and Active Compensation Analogy to reflex behavior in animals Bob Full and Berkeley collaborators –Reflex reactions to disturbances –< 100 ms –Relies on current state’s virtual elements At > 100 ms, active compensation is introduced –Cerebellum augments spinal reflexes –Corresponds to introduction of new virtual elements, new states to compensate

Contents Introduction Architectures for motion control Impedance control algorithms Explaining and reproducing human motion trajectories Disturbances - classification and strategies Conclusion

Results from Sagittal-plane open-loop dynamic optimization are encouraging –Improve –Implement similar horizontal-plane model Use results to derive closed-loop control laws Investigating use of Timed Model-Based Programming framework to address disturbance issues