Maciejewski’s paper was pretty tough for me to comprehend, though I get a much better “feel” of the numerical way of tackling the problem now.
Clearly, IK is not a linear problem, though we start off by using a gradient descent algorithm. Since the end effector’s end position & orientation (if it has both translational & rotational DOF) is a function of the DOF’s of all the joints in our skeleton, using a Jacobian matrix seems quite useful, wherein each entry refers to the effect on the end effector by changing a particular joint’s DOF.
The system could be over/under-constrained and pseudo-inverse Jacobian techniques are useful here; if the system were well-constrained, then the Jacobian inverse could be calculated. Both the above cases still suffer from degeneracy (when the derivative vectors line up). This is where SVD comes in. It allows us to decompose a matrix into a form that is easier to analyze and helps us know if it is singular/near-singular. The Jacobian transpose is also a cheap solution that compromises a bit on accuracy. The essence of numerical methods is in iteration and doing something until we deem it good enough.

The Footskate paper seemed to build on some of Mike’s previous work in Retargeting motions and used a really cool technique of manipulating bone length by moving the knee and ankles and the root a bit (w/o changing orientation) to prevent the sliding/slipping effect. Its also a separate post processing step and so it builds on other clean up mechanisms. It works on those frames with foot constraints (inferred automatically via previous work, though not the most accurate, thus requiring some manual editing). For the unconstrained frames, techniques like blending and displacement mapping are essential. I’m interested in knowing how these techniques don’t give rise to the footskate problem again (since they’re more of smoothing interpolation).

The Wei paper is the ” throw tons of data, infer stuff ” approach, allowing good deal of automation and coming up with pretty awesome results in real-time as well. The main disadvantage lies in the quality of data (importance of being non-redundant) though I feel this will be less of a concern if companies start sharing their mocap data. The main win here is that you don’t need to spend too much time in cleaning the data. I also think it can make use of previous runs to discard redundant data, making it more useful.

Analytical methods tackle simpler IK problems (or use assumptions that make it simpler) and less resource intensive and would be natural fit for games, though the data based approach also seems like something to build on. Numerical methods are cool to learn just because of the math and have a lot more power than the analytical methods. Data intensive methods which are parallelizable can be very useful both in speed and quality given a strong database and might be the favorite in the years to come.

The second paper I read was the footskate paper by Kovar. This paper gave an interesting method for solving the footskate problem by allowing for very slight changes in bone length. It seems to me like this would be extremely useful in applications of a more artistic nature, where just getting something that looks natural and “good” is the ideal. For a medical application where absolute fidelity is key, it perhaps might not be such a great idea to stretch bones.

The last paper I read was the Grochow paper. With no background in machine learning, this paper was a bit hard for me to parse. It seemed to me like they were getting good results performance-wise, but it was a little unclear to me how intuitive it would be to actually specify meaningful constraints and parameters in their system.

It was interesting to see the evolution of IK computation and processing over this series of papers. It seemed as if the early paper sought to highlight a key problem in IK (structure can be problematic), and suggested solutions that applied new constraints to the simplified models discussed in several of the earlier papers we’ve read this semester. The second paper attempted to resolve some IK issues by using computational techniques and the third attempted to resolve IK issues by supplementing raw data with machine learning/statistical techniques. The “evolutionary” pattern is also interesting in that a number of the papers read for Readings 0 of this course involving locomotion appear to still by operating on this kinematics-by-example idea discussed in the third paper.

The Kovar paper brought up an interesting approach to solving a very concrete perceptual problem that could be done approximately real-time, making it appropriate for relatively dynamic scenarios (such as crowd simulation as cited in the paper). The technique’s usefulness is also demonstrated on a real scenario in motion capture data. The second technique, while asserting a certain desire to be real-time, feels as if it should be more computationally expensive, not necessarily in terms of the complexity of the calculations but rather the volume of computations needed to adequately summarize and derive inferences from the training data. Also, it is not developed for conformity to motion capture data, so I would imagine that this particular technique could not be used in strictly choreographed scenarios.

The basic outline of the Maciejewski paper follows
the variations of using Jacobian based numerical
solutions to IK. The author explains how
ill-conditioned problems, primarily due to cases
where the Jacobian matrix is singular, can be
solved with minimal error by using the damped
least-squares method. He then explains how this
method can be applied to general dynamics problems
with good results.

I found the second paper harder to follow, but I
believe the basic idea was to develop closed form solutions to
IK problems in contrast to the numerical solutions
presented in the first paper. In addition, the
paper demonstrates a method for producing a
bipedal motion that maintains a proper posture
when the arms are given constraints. In some
respects, the use of a set of precomputed posture
‘stereotypes’ is similar to the third paper,
except the third paper pulls from a far larger
database of natural motion.

The third paper, by Wei, makes use of machine
learning techniques to distill a large database of
motions into a model that can be used to
intelligently fill in the gaps left by user
constraints. The author describes the purpose as a
system that allows novice animators to present a rough
sketch or design of a pose and the model
constructs what would be the most natural looking
pose to fit with the constraints. This paper was
of considerable interest to me as I attempted a
similar, though far cruder in comparison, project
in my neural networks class last year. I was
ultimately unsuccessful, but this paper
demonstrates that the general idea is sound.

Personally, I believe the roles of these methods
are directly related to the complexity of the
task. For the simplest models, maybe a few joints,
an analytical solution can be very effective. The
challenge of constructing said solution quickly
becomes prohibitive as the complexity of the
linkage grows, and thus numerical solvers become
more popular. But I think as we have become more
expectant of good, ‘real-looking’ motion, a more
all-encompassing approach is needed. The database
driven system can look at the entire human body at
once, generating a pose that works over its
entirety. The major drawback of course is that you
need large databases for the figure type (ie. bipedal
verses quadruped ) and the its slowness makes it
unsuitable for true real time applications.

-Real-Time IK for Anthropomorphic Limbs:
This paper did a great job of defining the issues that make solving IK in computer graphics difficult and the criteria for which IK methods should be evaluated. The authors determine that the main issues with CG IK are:
1)The amount of constraints to account for which can be simultaneous and conflicting.
2) Large # of degrees of freedom
3)Many joints are coupled (not independent and may move simultaneously)
4)Large amount of error
They go on to introduce an analytic/hybrid approach to IK and compare it with traditional numerical methods. Their method is faster and complete (in terms of solutions it accounts for) and far less unstable (it produces repeatable results which doesn’t happen numerically since the methods rely on initial guesses). Finally, they also suggest that the quality of IK methods should be based upon: efficiency, reliability, completeness, stability, and generality. Without a doubt, this paper provided the solution that Maciejewski’s circumvented; the method just seems to crush the purely numerical approach.

-Style-Based IK
Presents a method for using a probability distribution-based solution space for determining poses that follow a style learned from example data. The authors argue that most IK systems fail to keep solutions within a natural-looking space, making it so animators have to specify more constraints than necessary. Their method reduces this unnatural space by tending to choose solutions that closely resemble the training poses. I definitely see potential in this method; from my experience, rigging characters can be a pain in the ass, so the added constraint automation would alleviate some of this pain. Although, I do find the necessity for examples to be pretty limiting. Can you easily acquire the data to teach the intended style? And will the intended style always come through correctly? If the data is difficult to acquire and communicate effectively, then maybe it’s less trouble to just manually specify the constraints.

Although I’ve been informed that the numerical approaches are better suited for ill-formed constraints, it really seems like the analytic/hybrid approach is so much better. Maybe it’s the age of the Maciejewski paper, but the efficiency and repeatability of the analytic/hybrid would be so much better for any real-time application. Between the Numerical/Analytic approaches and the data approaches, it seems like it really just comes down to control versus speed. Obviously the data-driven approach is going to produce quality results quickly during production, but its diversity of movement is limited to the data you provide. On the other hand, although far more costly (in terms of time), numerical/analytic offers more control and consequently, more diversity. Using a database of millions of poses seems like it would be costly, redundant, and probably not dynamically usable in real-time, but it is needed in order to get diversity in pose solutions.

Thus, in terms of gaming, I can see data-driven examples being used for rapid development of animation sets for minor characters, while numerical/analytic being used for in-depth development of animation sets for main characters. Also, numerical/analytic would more likely be used in real-time computation due to efficiency and space (since a database of poses wouldn’t have to be loaded into memory or read from elsewhere).

This paper discusses the nature of the equations used for motion in articulated figures. In particular, the equations are ill-conditioned, meaning that no matter what you do, there is the potential that small perturbations in the input can cause large changes in the output. One tactic is to generate solutions using the pseudoinverse can result in discontinuous transitions. The preferred solution is using damped least-squares formulation which can be arbitrarily well-conditioned with the correct damping setting.

From what I gathered, it sounds like the psuedoinverse calculation is faster than the damped least-squares formulation. If that is the case, then pseudoinverse may be more appropriate for applications that demand speeds, such as games, while damped least-squares would be better for creating movies or other offline animation.

Footskate Cleanup for Motion Capture Editing

This paper proposes a post-processing method of removing footskate without introducing artifacts to the motion. This is achieved by making small changes to bone lengths in order to meet constraints without generating large changes in joint rotation. Since it is a post process, it doesn’t interfere with other algorithms, and allows other motion processing algorithms to introduce some footskate, knowing it will be removed in the post processing step.

Since the algorithm is post processing, it might not be suitable for real time application that demand a fast response. The paper suggested using it for characters in the backgrounds of games or crowed simulations, situations where the movements aren’t under intense scrutiny. Other methods, if faster, may be more applicable to conditions where speed is desired.

Natural Character Posing from a Large Motion Database

This paper proposes a method (nat-ik) for learning a prior for pose from a large motion database while maintaining interactive processing speed. This method is to be general over a human reachable space, unlike style-ik. The method deals with large databases by clustering poses and selecting representative frames from each cluster. FITC was used to approximate the Gaussian Process model, chosen for training speed, prediction speed, and error rate.

The impression I got was that that Nat-IK was a more general solution than style-ik, meaning style-ik might be more appropriate when a subset of motions is desired in higher accuracy.

The Maciejewski paper talked about some of the problems encountered when modeling simulations. It mentioned how these problems are often caused by the model itself rather than the equations. The Kallmann paper went into more inverse kinematics and how this can be used for animating specific contact or collisions more accurately. Finally, the Wu paper discussed methods for taking a subset of data from a motion database and using this creating more realistic inverse-kinematics. In general, it seems like a problem with inverse kinematics is that it can result in unrealistic motion between positions.

As for applications, it doesn’t seem like the ideas from the Wu paper would be as good for an interactive situation, compared to the other two papers. First, it needs to be trained on the data and then it still can take multiple seconds to generate a pose even after the training. Overall, inverse kinematics seems like something that would be useful, at least in some capacity, for most animation tasks.

* Footskate Cleanup for Motion Capture Editing by Lucas Kovar John Schreiner Michael Gleicher, SCA 2002.
This paper proposes a way to clean up a condition known as footskate (an artifact caused by feet moving when they are supposed to planted). The key idea of the paper is to not worry about preserving the length of the foot skeletons. By making the length of the flexible and considering the foot movement of the neighboring frames they manage to get a more realistic walking motion. I am surprised how much work had to be put in to solve a seeming minor issue.

*Natural Character Posing from a Large Motion Database. Wu, X; Tournier, M; IEEE CG&A November 2009.
This paper proposes a method of generating new natural looking poses from a large motion database. They have used kd-clustering to learn the priors from a large data base taking advantage of the fact that the poses are highly redundant. The algorithm is how ever limited to generating poses and doesn’t provide continuous motion and doesn’t work in real time.

For the Buss paper, I found the paper moderately dense, but it was a comprehensive overview which was necessary for understanding the basic approaches which are out there. However, the next two papers I read were more interesting and unique because they seemed to simplify the problem of IK without sacrificing too much in terms of accuracy and results. The Kovar paper was interesting because it actually suggested stretching as a solution, which I had not seen before, and it is innovative in its unorthodoxy.

The Style-Based IK paper was also interesting because of synthesizing poses, and methods to interpolate poses based on this method, which is something we talked about in class and may need to do for our projects. Also, it is just interesting because more than any other paper, it seems to try to abstract on data and move away from oversized data sets.

The first paper by Buss could be used for high quality animation software where the results are more important than real time behaviour. Whereas, Kallman work is suitable for low end animation systems or games where people get more focused on the context rather than contents of the animation. In such systems, having a fast and approximate model is more than sufficient.

I am quite skeptical of usefulness and effectiveness of “Millions of Example” paper. I think rules ( small number ) based systems could be better to mimic the humans varied actions ( a la Wolfram )