
Optimal Motion
Generation
Interpolation Methods based on
Learned Movements
Of all the remarkable physical abilities of humans, motor control
is the skill that is most often taken for granted, as it seems to
require the least conscious effort on our part. Only when a
particular motor skill is impaired or lost do we then begin to
appreciate the difficulty of the motor control and learning process.
It comes as no surprise that these exact same difficulties
are encountered when attempting to endow robots with motor
learning skills like that of humans.

Dancing motions generated for the KIBO humanoid robot 
Although we perform everyday motions quite effortlessly, it is all
too easy to forget that some of these motor skills are acquired
only after considerable effort, in some cases after years of practice.
From a child learning to throw a ball or swing a bat, to an adult
perfecting her golf swing or practicing calligraphy, or a gymnast
mastering a new tumbling routine, these seemingly quite distinct
motor skills become a part of our motion vocabulary only after years
of practice.
Yet we can can still infer some common sequence of events in the
motor learning process. Upon first learning the skill, the resulting
motion is usually crude and awkward, lacking grace and fluidity.
Initially it appears as if only a few degrees of freedom are
deliberately controlled, with the remaining degrees of freedom
either frozen in place or ``pulled along'' by mechanical and other
constraints. Then, as the motion is practiced, the oncelocked
degrees of freedom now seemingly begin to unlock, and move
in coordination with the primary degrees of freedom.
As the motion is perfected, all the degrees of freedom work
synchronously to create a fluid and natural looking motion. In
most cases the resulting motion is optimized in some sense with
respect to some physical criterion.
There is another common feature to the motor learning process:
the diminishing reliance on feedback as learning takes place.
Initially the motion will be guided and corrected by some form
of sensory feedback, e.g., visual or tactile. With practice,
the resulting motion appears more and more as if a preloaded
motor program is being played. In control theory terminology,
this transition can be described as moving from a closedloop
to an openloop control structure.
Clearly motor learning encompasses a much broader set of issues
than the ones touched upon above. There is, for example, the
complex interplay between the different modalities of perception
and action, as well as higher levels of ``physical" intelligence,
and the synthesis of ``balancing" and ``task" movement primitives.
Our current work in this area focuses on both optimal movement generation and interpolation methods based on
learned movements, as well as how elements of both methods can
be merged within a unified framework.


Optimal Movement Generation


There is ample biological evidence to justify an optimizationbased
approach to motor control and learning. Indeed, in the literature
one can find many optimal controlbased studies of various human
motions, e.g., maximumheight jumping, voluntary arm movements,
maintaining postural balance, minimumtime running and swimming,
even wheelchair propelling. Besides some of the more obvious optimization
criteria like minimum energy or control effort, strategies that involve
minimizing the jerk, as well as muscle or metabolic energy costs, have
also been examined in the context of specific arm motions.
Models for human motor learning and control that take into account
both the muscle dynamics and features of the neural system have been
proposed, and extensive studies of the cerebellum and its role in motor
learning and control have also been performed. The most predominant
models adopt a cybernetic metaphor, in which the body is treated as
a machine that receives sensory inputs and generates motor outputs.
Proponents of the equilibrium point hypothesis argue that motions
are generated according to a potential field determined by the
endpoints of the movement. Still others have argued that the
nervous system performs inverse dynamics to generate the motor
commands, with recent research suggesting that it is
possible to identify accurate internal models from movement data,
and that such strategies can be successfully implemented in robots.
More recent is the evidence suggesting that inverse dynamicsbased
feedforward computations are also an integral part of human movement
planning, particularly for fast movements.
From an engineering perspective an optimizationbased approach
to motion generation usually comes to mind as the first reasonable
thing to try. Past approaches have usually failed, however,
because the complexity of the governing equations of motion
usually led to intractable optimization problems. Indeed, the
intractability of the optimization seems at least partly (if
not largely) responsible for the recent flurry of attention
given to, e.g., neural networks, genetic algorithms, and
other evolutionary optimization approaches to motor learning.
One of the arguments that we put forth is that, like the view
of the cerebellum as a computational engine for dynamics,
modelbased motor learning based on classical descenttype
optimization methods is indeed a computationally feasible paradigm.
Aside from the complexity of the nonlinear dynamics, another
reason classical descent methods, despite their reliability
(indeed, in many cases these algorithms are the only ones that
can guarantee local optimality and convergence), are bypassed
in most motor learning schemes is their reliance on gradient and
Hessian information. Although in principle one can numerically
approximate these quantities, for problems involving even
moderately complex multibody systems, approximated gradients
and Hessians more often than not lead to illconditioning,
instability, and poor convergence behavior, not to mention
a significant increase in the computational burden.
What we have discovered in our research is that by appealing to
techniques from the theory of Lie groups, it is possible
to formulate the equations of motion of even complicated
antagonistic multibody systems like the human body in such
a way as to render the optimization problem tractable. In
many cases the solutions can even be obtained quite efficiently
and in a numerically robust way. We can, for example,
pose the active balancing problem as a secondorder
cone programming problem, for which efficient interior point
optimization algorithms are available. Human motions that
minimize muscle metabolic energy can also be reliably generated.
One of the keys lies in the ability to recursively evaluate the
nonlinear dynamics, and also to recursively compute exact analytic
gradients and Hessians without resorting to numerical approximations.
The resulting algorithms, while still computationintensive, are
O(n) with respect to the number of rigid bodies comprising the system. 





Balancing on a sliding floor 

Balancing on a spinning floor 





Minimum torque parallel bar motion 

Minimm torque rowing 





Minimum metabolic energy pedaling 

Minimum metabolic energy walking 
Related Publications
J.E. Bobrow, F.C. Park, and Athanasios Sideris,
``Progress on the Algorithmic Optimization of Robot Motion,"
in Fast Motions in Biomechanics and Robotics: Optimization and Feedback
Control, Diehl, Moritz; Mombaur, Katja (Eds.) , Lecture Notes in Control
and Information Sciences , Vol. 340, SpringerVerlag, October 2006.
Juyong Park, Jaeyoung Han, and F.C. Park, ``Convex optimization
algorithms for active balancing of humanoid robots," IEEE
Transactions on Robotics, Vol. 23, No. 4, pp. 817822, August 2007.
Sunghee Lee, Junggon Kim, F.C. Park, Munsang Kim and J.E. Bobrow,
``Newtontype algorithms for dynamicsbased robot movement optimization,"
IEEE Trans. Robotics Vol. 21, No. 4, pp. 657667, August 2005.
J. E. Bobrow, B. Martin, G. Sohl, E. C. Wang, Junggon Kim and F. C. Park,
``Optimal robot motions for physical criteria,"
J. Robotic Systems, Vol. 18, No. 12, pp. 785795, Dec 2001.
Albro, J.V., Sohl, G.A., Bobrow, J.E. and Park, F.C.,
``On the computation of optimal high dives,"
Proc. IEEE Conference on Robotics and Automation, San Francisco, 2000.



Interpolation Methods Based on Learned Movements


By now there is general agreement that humans learn a great variety
of movements, and that these movements are stored in some form in our
memories. What still remains unresolved is the exact nature of these
building blocks, or modules, for generating movements, and how these
modules are used to generate new movements. The idea that inverse
dyanamicsbased trajectories are stored in table form for every
possible movement has by now been dispelled on computational arguments.
On the other hand it has also become clear that the human movement
generation paradigm is fundamentally driven by the central nervous
system's remarkable capacity for storage and retrieval of large amounts
of data.
Our main motivation is to develop a method for generating natural
motions in an efficient manner: we seek to overcome the two main
disadvantages of dynamicsbased motion optimization, i.e., the
heavy computational burden, and the lack of modularity and
reusability. The innovative features of our approach are
(i) to select and classify a set of movement primitives, and
to store these as a family of basis functions; (ii) to extract
these basis functions from a factorization analysis of
observed human movements, and (iii) to perform higherlevel
processing (e.g., dynamicsbased motion optimization, construct
Hidden Markov model representations) using these basis functions
to generate natural motions in realtime.



Movement generation framework

In the block diagram describing our movement generation
framework, given a highlevel description of a desired task,
the task parser interprets this description and generates
a corresponding sequence of desired movements. The movement
compiler then takes this sequence as input and, drawing upon
the database of movement primitives, constructs an appropriate
movement sequence plan consisting of feedforward and feedback
terms to be applied over given epochs. The movement compiler
performs a combination of local optimization and learning,
motion interpolation, and motion blending and smoothing, all
the while ensuring that any physical and userspecified motion
constraints are satisfied.
The movement primitives are stored as a family of basis functions,
which are extracted from a factorization analysis (e.g., principal
component analysis, independent component analysis, or nonnegative
matrix factorization) of movement data. For example, arm motions
can be classified into reaching, swinging, lifting objects, and
various grasping and fine manipulation movements. Arbitrary
movements are then generated as linear combinations of the
basis functions such that they satisfy boundary conditions and
other feasibility criteria.






HMM designed and trained using human motion capture data to represent the specific arm motion.


The optimized arm motion that maximizes the probability for the trained HMM. In the optimization, we use the motion basis elements extracted from the PCA of motion capture joint angle trajectories.


HMMoptimized motion(left) versus torqueminimized motion(right). Both handraising motions are generated with the same boundary condition and motion basis elements(principal components). Note that the HMMoptimized motion appears more humanlike.

Related Publications
F.C. Park and Gyusang Jo, ;Movement primitives and principal component analysis," Advances in Robot Kinematics (J. Lenarcic and C. Galletti, Eds.), Kluwer, Dordrecht, 2004
Bokman Lim, Syungkwon Ra, and F.C. Park, "Movement primitives, principal component analysis, and the efficient generation of natural motions" , in Proc. IEEE Conference on Robotics and Automation, Barcelona, 2005
Junghyun Kwon and F. C. Park, "Using hidden Markov models to generate natural humanoid movement," in Proc. IEEE/RSJ International Conference on Intelligent Robots and Systems, Beijing, 2006


