Dbanks Matthale

Note on Reuse of Content

Several portions of this assignment are taken, in whole or in part, from Matthew Hale's C.2 assignment. To avoid self-plagiarism, this and other disclaimers at the beginning of each such section have been included.

Desired Behavior or Capability

Portions of this section are taken from Matthew Hale's C.2 assignment.
Among wild animals, the diversity of terrains and environments has promoted, by means of evolution, development in animals of myriad behaviors which help the animals survive and, ideally, thrive in such environments. Among the many capabilities of animals, their ability to locomote is of paramount importance, as it enables them to flee predators, chase prey, find water or shelter, and to otherwise interact with their surroundings. While a robot in the same environment as an animal does not necessarily have to avoid predators or find resources the same way that an animal does, a robot's efficacy in a particular environment is frequently judged based upon its ability to match (and ideally, supersede) an animal's ability to locomote in the same environment.

While an animal may use several methods of locomotion to respond to the demands of different terrains in their environment, there exists a family of gaits within each mode of locomotion. For example, animals which locomote over flat ground may use several different gaits to produce stable locomotion. Research into animal locomotion suggests that transitions between the various gaits which may be used on a given terrain are made in order to maintain efficient locomotion as an animal changes speed [2]. The problem which we wish to address is to transition stably between efficient gaits in a robot as the robot changes speeds. More specifically, a the user commands the robot to change speeds, we want to change to not only a new speed, but also change to a completely new gait which is optimal (or nearly optimal; this point is expanded upon further) for that speed, and to make this transition in a stable manner. To isolate the variable of stability, we will examine gaits over a "homogeneous" surface. While real-world terrains are not perfectly homogeneous, we will consider a terrain which is homogeneous enough that any instability which arises in the robot's locomotion is definitely the result of our transitions and not a product of the terrain.

The value of gait transitions is wide, varied, and great (Matthew Hale's C.2 elaborates on this point). Here, the ability of gait transitions based upon commanded speed changes is valuable in that it allows the robot to waste less energy while locomoting by tailoring efficiency around speed. Being able to change the robot's entire gait (i.e., all of its runtime parameters) as the speed of the robot is changed enables the robot to conform its gait at each speed to the task of minimizing the energy required to use that gait. In order to change the robot's entire gait to take advantage of using minimal energy across a whole family of gaits (in our case, a set of gaits meant for use on homogeneous terrain), there must be stable gait transitions between such gaits. Thus, the value of our project is that it enables the user to take advantage of efficiency across a wide range of speeds.

Present Unavailability

Among the literature identified in Matthew Hale's C.2 assignment, there does not appear to be any influence had by speed changes on gait generation. [11] makes use of different gaits (which in the context of this paper are footfall patterns wherein each individual footfall is planned) on the Honda ASIMO robot in order to avoid obstacles as the robot moves. [14] focuses on transitioning between wheeled and legged gaits (because the robot has both legs and wheels) as a means of responding to changing environmental demands. [1] uses different gaits on the RHex robot which are meant to handle walking on flat ground to climbing stairs and on the RiSE robot to climb vertically, turn the robot while it is climbing, and move horizontally across a vertical surface. And [15] uses different gaits as a way of expanding the capabilities of the RiSE robot to be able to traverse horizontal surfaces, vertical surfaces, and the interfaces between them. Thus while roboticists have made extensive use of a wide array of different gaits, there does not appear to have been a major exploration of gait transition capabilities motivated by maintaining efficiency while changing speeds.

In addition, individual gaits (i.e., gaits at single speeds) have been tuned to minimize their energetic cost [3], meaning that there has certainly been focus on generating optimized gaits in the past. However, there does not appear to be any robotics literature which focuses on combining the task of transitioning between gaits with the task of minimizing the energetic cost of locomotion.

Note on Unavailability

While it certainly would have been more convincing to find a direct quote discussing the absence of gait transitions driven by speed modulation, no such quote could be found. We attribute this to the great poverty of literature on gait transitions. What papers are out there generally discuss the applications of gait transitions and use gait transitions to have a robot locomote between two very different terrains. As a result, we could not find any discussion of speed modulation-driven gait transitions; this lack of discussion and our examination of research papers pertaining to the subject of gait transitions leads us to believe that this problem has not yet been addressed and that it is therefore appropriate for this assignment.

Desirability of Bioinspiration

Animals have the remarkable ability to make stable transitions between speeds, and biological research suggests that animals such make gait transitions as they change speeds in order to minimize the energy expended at that speed. In particular, [2] showed that horses change from a trot to a gallop in order to minimize the energetic cost of their locomotion. Furthermore, the authors of [7] remark that the energetically optimized running gaits (tolt, trot, pace, left and right canters and gallops) could be considered a "kinematic continuum", in which the gait parameters change continuously rather than discretely as speed increases. It seems possible that by recreating such a continuum in a legged robot, the robot could achieve stable speed transitions using gaits along this continuum.

The Hypothesis

From the observation that animals optimize their walking and running gaits based on efficiency [2], in addition to the fact that for at least some range of speeds these gaits form something of a "kinematic continuum [7]," we hypothesize that stable speed transitions in legged robots can be made by traversing an interpolated, continuous path through gaitspace along gaits of optimal efficiency. We further hypothesize that this method of gait transition will allow for more rapid speed transition than is stably possible by merely adjusting the period of the gait.

The Idea

We will first optimize gaits for efficiency (as quantified by specific resistance; specific resistance will be used as it was originally devised in [16] as a means of identifying the cost of travel and was used by [3] to optimize gaits successfully on a different platform) at several different speeds, sampling gaitspace in an algorithmic manner using Nelder-Mead descent to find optimal gaits at several fixed speeds. We will then interpolate these data in order to produce a continuum of efficient gaits at different speeds. While ensuring that each gait which we generate is optimized for the speed at which it runs, we will operate under the assumption that the gaits generated by interpolation represent reasonably optimal gaits for each speed.
Secondly, we will perform speed transitions moving along this gait continuum. This will take the form of a simple experiment wherein we will gradually increase and decrease the speed of the robot while morphing the Buehler clock based on our interpolated family of optimal gaits. The test of our hypothesis is whether or not the robot falls while traversing this continuum. We have a binary conception of stability, which is to say that we will judge robot motion and transitions to be stable as long as the robot does not hit its body on the ground (as was done to judge stability in Lab 1).
If the robot is able to make stable "slow" speed transitions (i.e., transitions in which the robot is given time to fully transition to a higher-speed gait before the speed is further increased), we will then compare its ability to make rapid transitions in this manner with its ability to make rapid transitions when naively adjusting the gait period. This will be accomplished by choosing a starting speed and an ending speed and then transitioning speeds using our method in addition to transitioning speeds using the naive gait period adjustment. The means of comparison of stability between the two methods of changing gaits will be in finding which means of speed transitioning causes the robot to go unstable first.


If traversing the path of optimally efficient gaits is shown to be an unstable means of gait transition, we can say that the first part of our hypothesis was incorrect. However, in this case the research could still be useful if we can determine piecewise-continuous regions of gaitspace along which stable transitions can be made, because our hypothesis may still hold true in such regions. For instance, there may exist "walking" and "running" regimes characterized by fundamentally different gaits such that stable transitions via the path of efficient gaits between these two regions is not possible. However, in such a case it may still be possible to transition within the confines of each region using our prescribed method. This dynamic instability arising from the abrupt regime transition would likely occur a certain critical speed. Programming a special transition behavior between valleys that overlap over a certain speed interval could perhaps overcome this particular problem, though this problem does decidedly present a refutation of the hypothesis in its present state.

Additionally, it is possible that the first part of the hypothesis holds true but the second does not, which would be demonstrated by showing that merely adjusting the leg-frequency parameter of a gait (namely the parameter s) can produce faster stable speed transitions than are produced by transitioning along the path of efficient gaits. It is possible that the path of gaits which are efficient with respect to specific resistance does not conduce to rapid parameter changes. We may find that changing all of the robot's runtime parameters simultaneously causes instability, while leaving every parameter except leg speed fixed at a constant value promotes stability as fewer changes need to be made while locomoting. Even in such an instance, we could potentially still make use of our interpolated family of gaits as we could stagger the parameter changes (so that they do not all change at once), or else consider clustering parameters over a few speeds so that a small range of speeds would use a single set of fixed gait parameters. This would make use of our notion of where efficient gaits lie in gaitspace by making some gait transitions, while simultaneously exploiting the stability of scaling only the speed parameter by doing so over small regions. In this way, the problem might be remedied by naively scaling only leg speed over small partitions of gaitspace while intelligently modifying all gait parameters at the interfaces between these partitions.

Necessary Means

The necessary tools to complete our experiment are:
1. A robot which is able to run many different gaits defined by different parameter sets (which in turn define different Buehler clocks)
2. A cost function which can normalize the cost of a gait over different speeds (we intend to use specific resistance as it is used by [3] to optimize gaits).
3. A tuning algorithm which coordinates changing the robot's gait parameters with values of the cost function in order to find gait parameters which minimize costs.
4. A means of storing the gaits generated by the tuning algorithm
5. A means of mathematically interpolating the gaits stored in order to produce a continuum of efficient gaits
6. A suitably long stretch of (roughly) homogeneous terrain on which to test the gaits and their transitions
7. The existing, naive set of gaits to serve as a comparison

The following discussion of the Robotics Authority and Biology Authority and their respective precursor and successor papers is taken partially from Matthew Hale's C.2 with significant additions here to correspond to this assignment. In addition, portions of the discussion on the credentials of these papers and their respective authors is taken partially from Matthew Hale's C.1 assignment.

Robotics Literature

The major source of authority we are considering from robotics is [1]. In this paper, the authors create and then transition between very different gaits which are used on very different terrains. On the RHex platform, the authors switch between an alternating tripod gait on flat ground and a metachronal gait on stairs. On the RiSE platform, the authors switch between an alternating tripod, a tetrapod, and a pentapod gait in order to climb a vertical surface, turn the robot while climbing, and to move horizontally across a vertical surface. This paper represents important contributions to robotics as the authors adapt two robotic platforms to handle not only different gaits (and in one instance, a completely different terrain), but also to handle the interfaces between these gaits in gaitspace. This is very important as it enables these robots to switch between these behaviors in realtime, making them usable on the fly. We feel that this paper is an important source of authority based on its venue and authors. This paper came from the IEEE International Conference on Robotics and Automation (ICRA). ICRA is an annual, international conference which is hosted by the IEEE's Robotics and Automation Society. According to the website for this organization:

The [Robotics and Automation] Society strives to advance innovation, education, and fundamental and applied research in Robotics and Automation.

It is clear that the research-centric nature of this conference is in line with what we seek to find here. Further, the society which hosts this conference is a subsidiary organization of the IEEE. The IEEE is one of the world's most respected engineering organizations and greatly contributes to the credentials of this conference. Based on the stated goals of the organization hosting ICRA and based on the fact that this organization is a part of the IEEE, this conference appears to be very credible and thus a good venue in which to find a basis for this assignment. In addition, the second author for this paper (who is the Principal Investigator) is now listed as Adjunct Faculty at the Robotics Institute at Carnegie Mellon as well as Head Robot Scientist at Boston Dynamics. As Dr. Rizzi holds positions at both a well-respected research university and one of the leaders in industry in robotics research, he gives significant credibility to this paper. The first author of this paper is now a Postdoctoral Researcher at the University of Pennsylvania for Professor Dan Koditschek. Given Dr. Haynes position at a well-known research university and considering that he works directly with Professor Koditschek who has an extensive research career which includes several landmark contributions to robotics, Dr. Haynes's present appointment suggests that he, too, is a credible author. We felt that this paper was worth tackling for this assignment because it represents one of the major papers emphasizing gait transitions in legged robots and was the source of inspiration for much of this assignment.

Precursor Literature

The authors of this paper cite [5] several times throughout their paper as they discuss various gaits, which suggests to us that this is an important "neighbor" paper. Google Scholar shows that this paper has been cited 64 times, meaning that it has been judged to be important by many authors other than those on [1]. This paper is relevant because it explores the stair climbing behavior in RHex, which is a key part of [1]. In [1], the authors experiment with gait transitions between walking and stair climbing with RHex, meaning that [5] creates a clear context for [1] because we are able to see how gaits were developed for RHex and how the need for gait transitions was addressed thereafter. This paper relates to our topic because it explores different gaits within a single environment (and thus a single set of constraints), which is what we intend to do, albeit with a more rigorous, transition-driven approach.

The second predecessor paper we have chosen is [3]. This paper focuses on using an automated method (specifically, using the Nelder-Mead algorithm to minimize specific resistance) to generate gaits on the RHex platform. This was chosen to be a good "neighbor" paper for [1] because it represents a means by which the robots in [1], RHex and RiSE, or seemingly any other platform, can be made to behave more efficiently. In particular, [3] is relevant to [1] because it enables more gaits on both RHex and RiSE and thus requires transitions between these new gaits. [3] is relevant to our project because it provides a clear, replicable starting point for us, i.e., we can begin by generating optimized gaits in gaitspace by using the methods outlined in [3] and then interpolate these data to generate the family of gaits we desire.

Successor Literature

The first successor paper we are considering is [15]. Google Scholar shows that this paper came out in 2008, but already has 38 citations, making it the most cited successor paper to [1] in spite of its youth. This paper is relevant because it further explores locomotion on the robotic RiSE platform and mentions in particular (in its abstract) that the authors are considering locomotion on both flat ground and on a vertical surface. This paper is a very good successor paper as it further expands the gait transitions of the RiSE robot beyond those discussed in [1]. This relates to our topic in that it motivates the need for efficiency within individual terrains. As robots become more capable and are able to tackle more terrains, there will arise a constant need to be efficient and stable not only at the interfaces between these terrains, but also within these terrains as the robot's change behaviors or speeds. This paper motivates our topic in that we seek to address the need for efficiency while terrains are constant, a need which this paper does not address and thus leaves open.

The second successor paper we are considering is [14]. This paper was released only 4 months ago, so it does not yet have any citations. However, it is a good successor paper because it considers gait transitions in ways beyond those explored in [1]; in particular, [14] considers a platform which can switch between both legged and wheeled methods of locomotion. Within the legged capabilities of their platform, the authors discuss a means of minimizing energy use, which is also related to earlier biology research. I feel that this is a good "neighbor" paper because it presents a robot which can not only transition among legged gaits, but which can also change its means of locomotion entirely, thus providing a broad extension of the type of gait transitions explored in [1]. This provides further motivation for gait transitions between distinctly different methods of locomotion. In addition, the attention given to minimizing energy use in certain legged behaviors on this platform (the authors mention specifically minimizing energy use while standing up) relates to our project in that we seek to minimize energy over a broad range of speeds while locomoting. This paper also motivates our idea in that as different terrains are conquered (which is one of the stated goals for the PEOPLER-II robot used in this paper), the need for efficiency within those terrains becomes more necessary. We, as mentioned above, are attempting to begin to fill in the need for efficient locomotion while moving over a constant terrain.

Biological Authority

Our source of biological authority is [2]. In [2], the authors are concerned with what causes horses to transition from a trot gait to a gallop gait; they focus on whether the impetus for change lies in the musculoskeletal forces experienced by the horse, or whether it is the result of metabolic considerations. This study runs horses in two environments which have the same musculoskeletal forces, but different metabolic factors and concludes that gait transitions are the result of metabolic efficiency. There are four authors listed on this paper, all of whom hold or have held positions at Caltech. The first author of the paper, Steven J. Wickler, passed away in 2007. During his research career, he was the director of the Equine Research Facility at Caltech and the director of Laboratory Animal Facilities at Caltech; the fact that Dr. Wickler held two leadership research positions at a well-respected university such as Caltech adds to his credibility. Google Scholar lists his h-index as 17, which is a respectable figure. In addition, this paper is from the Journal of Experimental Biology. According to the ISI Journal Citation Reports, the Journal of Experimental Biology is ranked 19th out of 76 in terms of impact factor, with an impact factor of 2.722. This puts the Journal of Experimental Biology just inside the upper one fourth of biology journals, which is a respectable rank. However, in terms of total citations, the Journal of Experimental Biology has 20334, which puts it ranked 4th out of 76 by ISI. This is a very high rank and indicates that the Journal of Experimental Biology has great influence on other researchers. The combined credentials of this paper's authors and venue lead us to believe that it is a valid source of authority for a topic such as ours.

Predecessor Literature

The first precursor paper which we are considering is [18]. This paper is a good "neighbor" paper to [2] because it performed a low-level study of the mechanisms which determine gaits in turkeys, namely leg muscles and tendons, and found that these mechanisms adjust themselves to store and then release energy when it is most beneficial. In broader terms, turkeys were found to constantly adjust their gait to use less energy, regardless of the speed at which they were moving. This relates to [2] as it provided evidence of gait transitions as a means of saving energy within an animal other than horses. This relates to our topic as it provides further bio-inspiration for gait transitions according to efficiency in robots. In addition, it gives more credibility to our hypothesis because it shows that, while our main biological source studied horses, the benefits had by changing gaits while changing speeds are not exclusive to horses, nor to quadrupeds.

The second precursor paper we are considering is [19]. This paper is a good "neighbor" paper as it appears to present one of the first arguments that it is global metabolic economy of an animal (i.e., total energy used by the animal) which drives gait transitions, not local metabolic economy (i.e., the energy used by each leg). This paper represents a very strong foundation upon which [2] and other papers concerned with gait transitions are based. It provides a very important background for [2] as [2] examines a particular segment of gaitspace wherein transitions are made according to global metabolic economy. This is very relevant to our topic as it suggests that we should optimize gaits for global robot energy efficiency. We state above that we intend to do this when we state that we intend to use specific resistance, and this paper bolsters the notion that this method will yield positive results.

Successor Literature

The first successor paper we are considering is [6]. This paper is relevant to [2] because it considers mechanical and metabolic details of gaits in ostriches, which are bipeds, while [2] studies the same aspects of gaits but instead on horses, which are quadrupeds. [6] concludes that ostriches select their gaits at different speeds (in the paper ranging from 0.8 m/s to 6.7 m/s) in order to minimize energy use at that speed, which is the same conclusion reached in [2] about horses. This paper is relevant to our topic as it bolsters the notion that there are efficiency advantages to be gained using gait transitions by very different animals in the animal kingdom. The fact that very different animals (thus far, horses, turkeys, and ostrichers) transition gaits over a range of speeds (and in this case, over a very wide range of speeds) to save energy leads us to believe that robots stand to benefit in the same way by using the same behavior.

The second successor paper that we will consider is [17]. As the title suggests, a key part of this paper is relating the motion of animals to the energy associated with that motion and mapping these characteristics of a gait to gait parameters which define it. We felt that this was a logical choice of "neighbor" paper because it directly expanded upon the work done in [2]. In particular, [2] focuses on a single gait transition in horses. [17] builds upon this work and considers several transitions which occur in several different species and contrasts the characteristics of each of these transitions. This paper is relevant to our project not only because it explores gait transitions, but also because it seeks a (preliminary) probing of the gaitspace of humans and horses. This paper finds that there do appear to be discontinuities in gaitspace between gaits which are qualitatively "walks" versus those which are qualitatively "runs" and that this continuity is not sufficiently explained by changes in duty factor. While we hypothesize that we can find a smooth path through gaitspace, this paper provides evidence that we may not find this path, where discontinuities lie, and rules out one possible explanation for why discontinuities may occur.

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3. Joel D. Weingarten, Gabriel A. D. Lopes, Martin Buehler, Richard E. Groff, Daniel E. Koditschek,
"Automated Gait Adaptation for Legged Robots", IEEE International Conference on Robotics Automation (ICRA), April 2004
4. Ohung Kwon and Jong Hyeon Park, "Gait Transitions for Walking and Running of Biped Robots", IEEE International Conference on Robotics Automation (ICRA), September 2003
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