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As humans move through their environment, they must change the [[stiffness]] of their joints in order to effectively interact with their surroundings. Stiffness is the degree to a which an object resists deformation when subjected to a known force. This idea is also referred to as impedance, however, sometimes the idea of deformation under a given load is discussed under the term "compliance" which is the opposite of stiffness (defined as the amount an object deforms under a certain known load).
In order to effectively interact with their environment, humans must adjust the stiffness of their limbs. This is accomplished via the co-contraction of antagonistic muscle groups.<ref name= Hogan>{{cite journal|last=Hogan|first=Neville|title=The Mechanics of Multi-joint Posture and Movement Control|journal=Biological Cybernetics|year=1985|volume=52|issue=5|pages=315–331}}</ref><ref name= VanHam>{{cite journal|last=Van Ham|first=R.|coauthors=Sugar, T.G.; Vanderborght, B.; Hollander, K.W.;Lefeber, D.|title=Compliant Actuator Designs|journal=Robotics & Automation Magazine, IEEE|year=2009|volume=16|issue=3|pages=81–94}}</ref>

Humans use neural control along with the mechanical constraints of the body to adjust this stiffness as the body performs various tasks. It has been shown that humans change the stiffness of their limbs as they perform tasks such as hopping,<ref name= Oliver>{{cite journal|last=Oliver|first=J.L.|coauthors=Smith, P.M.|title=Neural Control of Leg Stiffness During Hopping in Boys and Men|journal=Journal of Electromyography and Kinesiology|year=2010|volume=20|issue=5|pages=973–979}}</ref> performing accurate reaching tasks,<ref name= Lametti>{{cite journal|last=Lametti|first=Daniel R.|coauthors=Houle, Guillaume; Ostry, David J.|title=Control of Movement Variability and the Regulation of Limb Impedance|journal=Journal of Neurophysiology|year=2007|volume=98|issue=6|pages=3516–3524}}</ref> or running on different surfaces.<ref name= Ferris>{{cite journal|last=Ferris|first=Daniel P.|coauthors=Louie, Micky; Farley, Claire T.|title=Running in the Real World: Adjusting Leg Stiffness for Different Surfaces|journal=Proceedings of the Royal Society of London. Series B: Biological Sciences|year=1998|volume=265|issue=1400|pages=989–994}}</ref>

While the exact method by which this neural-modulation of limb stiffness occurs is unknown, many different hypotheses have been proposed. A thorough understanding of how and why the brain controls limb stiffness could lead to improvements in many robotic technologies that attempt to mimic human movement.<ref name="VanHam"/>

== Stiffness ==

	Stiffness is typically viewed as a material property describing the amount a material deforms under a given force as described by [[Hooke's law]]. This means that objects with higher stiffness are more difficult to bend or deform than objects with lower stiffnesses. This concept can be extended to the limbs and joints of biological organisms in which stiffness describes the degree to which a limb or joint deflects (or bends) under a given load. Limb stiffness can also be described as the static component of [[Mechanical impedance|impedance]].<ref name=Hogan /><ref name= Trumbower>{{cite journal|last=Trumbower|first=Randy D|coauthors=Krutky, M.A.; Yang, B.; Perreault, E.J.|title=Use of Self-Selected Postures to Regulate Multi-Joint Stiffness During Unconstrained Tasks|journal=PloS one|year=2009|volume=4|issue=5|pages=e5411}}</ref> Humans change the stiffness of their limbs and joints to adapt to their environment.<ref name=Ferris />  Limb and joint stiffness has been previously studied and can be quantified in various ways. The basic principle for calculating stiffness is dividing the deformation of a limb by the force applied to the limb, however, there are multiple methods of quantifying limb and joint stiffness with various pros and cons. It is interesting to note that when quantifying limb stiffness, one cannot simply sum the individual joint stiffnesses due to the nonlinearities of the multi-joint system.

A few of the specific methods for calculating limb stiffness can be seen below:<ref name= Butler>{{cite journal|last=Butler|first=R.J.|coauthors=Crowell, H.P.; Davis, I.M.|title=Lower Extremity Stiffness: Implications for Performance and Injury|journal=Clinical Biomechanics|year=2003|volume=18|issue=6|pages=511–517}}</ref>

Vertical Stiffness (''k''<sub> ''vert''</sub>) is a quantitative measure of leg stiffness that can be defined by the equations below:<ref name=Butler />

 <math>K_\text{vert}=\frac{F_\text{max}}{\Delta y}</math>

Where ''F''<sub> ''max''</sub> is the maximum vertical force and delta y is the maximum vertical displacement of the center of mass 
 <math>K_\text{vert}=m(\frac{2\pi}{P})^2</math>

Where m is the body mass and P is the period of vertical vibration 
  <math>K_\text{vert}=m\omega_\text{0}^2</math>

Where m is the mass of the body mass and ''&omega;''<sub> ''0''</sub> is the natural frequency of [[oscillation]]

Limb Stiffness (K_limb) is the stiffness of the entire limb and can be defined by the equations below:

 <math>K_\text{limb}=\frac{F_\text{max}}{\Delta L}</math>

Where ''F''<sub> ''max''</sub> is the maximum applied force and ''ΔL'' is the change in length of the limb

[[Torsion (mechanics)|Torsional]] Stiffness (K_joint)is the rotational stiffness of the joint and can be defined by the equations below:

 <math>K_\text{joint}=\frac{\Delta M}{\Delta \theta}</math>

Where ''ΔM'' is the change in joint moment and ''Δ&theta;'' is the change in joint angle
 <math>K_\text{joint}=\frac{2W}{\Delta \theta}</math>

Where W is the negative mechanical work at the joint and ''Δ&theta;'' is the change in joint angle

These different mathematical definitions of limb stiffness help to describe limb stiffness and show the methods by which such a limb characteristic can be quantified.

== Stiffness modulation ==
The human body is able to modulate its limb stiffnesses through various mechanisms with the goal of more effectively interacting with its environment. The body varies the stiffness of its limbs by three primary mechanisms: muscle cocontraction,<ref name=Hogan /><ref name= Ludvig>{{cite journal|last=Ludvig|first=Daniel P|coauthors=Kearney, R.E.|title=Real-time estimatoin of intrinsic and reflex stiffness|journal=Biomedical Engineering, IEEE|year=2007|volume=54|issue=10|pages=1875–1884}}</ref> posture selection,<ref name=Trumbower /> and through stretch reflexes.<ref name=Hogan /><ref name= Nichols>{{cite journal|last=Nichols|first=T.R|coauthors=Houk, J.C.|title=Improvement in linearity and regulation of stiffness that results from actions of stretch reflex|journal=J. Neurophysiology|year=1976|volume=39|issue=1|pages=119–142|pmid=1249597}}</ref><ref name= Shemmell>{{cite journal|last=Shemmell|first=Jonathan|coauthors=Krutky, M.A.; Perreault, E.J.|title=Stretch sensitive reflexes as an adaptive mechanism for maintaining limb stability|journal=Clinical Neurophysiology|year=2010|volume=121|issue=10|pages=1680–1689|pmid=20434396|pmc=}}</ref> <ref name= Trumbower_Finley>{{cite journal|last=Trumbower|first=RD|coauthors=Finley, J.M.; Shemmell, J.B.; Honeycutt, C.F.; Perreault, E.J.|title=Bilateral impairments in task-dependent modulation of the long-latency stretch reflex following stroke|journal=Clinical Neurophysiology|year=2013|volume=124|issue=7|pages=1373–1380|pmid=23453250}}</ref>

Muscle cocontraction (similar to [[muscle tone]]) is able to vary the stiffness of a joint by the action of antagonistic muscles acting on the joint. The stronger the forces of the antagonistic muscles on the joint are, the stiffer the joint becomes.<ref name=VanHam /><ref name=Ludvig /> The selection of body posture also affects the stiffness of the limb. By adjusting the orientation of the limb, the inherent stiffness of the limb can be manipulated.<ref name=Trumbower /> Additionally, the stretch reflexes within a limb can affect the stiffness of the limb, however these commands are not sent from the brain.<ref name=Nichols /><ref name=Shemmell />

=== Locomotion and hopping ===
As humans walk or run across different surfaces, they adjust the stiffness of their limbs to maintain similar locomotor mechanics independent of the surface. As the stiffness of a surface changes, humans adapt by changing their limb stiffness. This stiffness modulation allows for running and walking with similar mechanics regardless of the surface, therefore allowing humans to better interact and adapt with their environment.<ref name=Oliver /><ref name=Ferris /> The modulation of stiffness therefore, has applications in the areas of [[motor control]] and other areas pertaining to the neural control of movement.

Studies also show that the variation of limb stiffness is important when hopping, and that different people may control this stiffness variation in different ways. One study showed that adults had more feedforward  neural control, muscle reflexes, and higher relative leg stiffness than their juvenile counterparts when performing a hopping task. This indicates that the control of stiffness may vary from person to person.<ref name=Oliver />

=== Movement accuracy ===
The [[nervous system]] also controls limb stiffness to modulate the degree of accuracy that is required for a given task. For example, the accuracy required to grab a cup off of a table if very different than that of a surgeon performing a precise task with a [[scalpel]]. To accomplish these tasks with varying degrees of required accuracy, the nervous system adjusts limb stiffness.<ref name= Lametti/><ref name= Trumbower/> To accomplish very accurate tasks higher stiffness is required, however, when performing tasks where accuracy is not as imperative, lower limb stiffness is needed.<ref name= Lametti/><ref name= Trumbower/> In the case of accurate movements, the central nervous system is able to precisely control limb stiffness to limit movement variability. The [[cerebellum]] also plays a large role in controlling for the accuracy of movements.<ref name="purves book"/>

This is an important concept for everyday tasks such as tool use.<ref name= Trumbower/><ref name= Burdet>{{cite journal|last=Burdet|first=E.|coauthors=Osu, R.; Franklin, D.W.; Milner, T.E.; Kawato, M.|title=The central nervous system stabilizes unstable dynamics by learning optimal impedance|journal=Nature|year=2001|volume=414|issue=6862|pages=446–449|pmid=11719805}}</ref> For example, when using a screwdriver, if limb stiffness is too low, the user will not have enough control over the screwdriver to drive a screw. Because of this, the central nervous system increases limb stiffness to allow the user to accurately maneuver the tool and perform a task.

== Neural control ==
The exact mechanism for the neural control of stiffness is unknown, but progress has been made in the field with multiple proposed models of how stiffness modulation may be accomplished by the nervous system. Limb stiffness has multiple components that must be  controlled to produce the appropriate limb stiffness.

=== Combination of mechanics and neural control ===

Both the neural control and the mechanics of the limb contribute to its overall stiffness. The cocontraction of antagonistic muscles, posture of the limb, and stretch reflexes within the limb all contribute to stiffness and are affected by the nervous system.<ref name=Hogan /><ref name=Trumbower />

The stiffness of a limb is dependent on its configuration or joint arrangement.<ref name=Hogan /><ref name=Trumbower /> For example, an arm that is slightly bent, it will deform more easily under a force directed from the hand to the shoulder than an arm that is straight. In this way, the stiffness of a limb is partially dictated by the limb's posture. This component of limb stiffness is due to the mechanics of the limb and is controlled voluntarily.

=== Voluntary vs. involuntary stiffness modulation ===

Some components of limb stiffness are under voluntary control while others are involuntary.<ref name=Trumbower /> The determining factor as to whether a component of stiffness is controlled voluntarily or involuntarily is the timescale of that particular component's method of action. For example, stiffness corrections that happen very quickly (80-100 milliseconds) are involuntary while slower stiffness corrections and adjustments are under voluntary control. Many of the voluntary stiffness adjustments are controlled by the [[motor cortex]] while involuntary adjustments can be controlled by [[reflex]] loops in the [[spinal cord]] or other parts of the brain.<ref name=Ludvig /><ref name=Nichols /> <ref name="purves book">{{cite book|editor=Dale Purves et al. |title=Neuroscience.|year=2007|publisher=W. H. Freeman|location=New York|isbn=0878936971|edition=4th }}</ref>

Stiffness adjustments due to reflexes are involuntary and are controlled by the spinal cord while posture selection is controlled voluntarily.<ref name=Trumbower /> However, not each component of stiffness is strictly voluntary or involuntary.<ref name=Ludvig /> For example, Antagonistic muscle cocontraction can be either voluntary or involuntary. Additionally, because much of the legs' movements are controlled by the spinal cord and because of the larger neural delay associated with sending signals to the leg muscles, leg stiffness is more involuntarily controlled than arm stiffness.

=== Possible neural control models ===

Researchers have begun implementing controllers in into robots to control for stiffness. One such model adjusts for stiffness during robotic locomotion by virtually cocontracting antagonistic muscles about the robot's joints to modulate stiffness while a [[central pattern generator]] (CPG) controls the robot's locomotion.<ref name= Xiong>{{cite journal|last=Xiong|first=Xiaofeng|coauthors=Worgotter, F.; Manoonpong, P.|title=An adaptive neuromechanical model for muscle impedance modulations of legged robots}}</ref>

Other models of the neural modulation of stiffness include a [[Feed forward (control)|feedforward]] model of stiffness adjustment. These models of neural control support the idea that humans use a feedforward mechanism of stiffness selection in anticipation of the required stiffness needed to accomplish a given task.<ref name= Hu>{{cite journal|last=Hu|first=Xiao|coauthors=Murray, W.M.; Perreault, E.J.|title=Biomechanical constraints on the feedforward regulation of endpoint stiffness|journal=Journal of Neurophysiology|year=2012|volume=108|issue=8|pages=2083–2091|pmid=22832565|pmc=}}</ref>

Most models of the neural control of stiffness promote the idea that humans choose an optimal limb stiffness based on their environment or the task at hand. Studies postulate the humans do this in order to stabilize unstable [[Dynamics (mechanics)|dynamics]] of the environment and also to maximize the energy efficiency of a given movement.<ref name=Trumbower /><ref name=Burdet /> The exact method by which humans accomplish this is unknown, but impedance control has been used to give insight into how humans may choose an appropriate stiffness in different environments and as they perform different tasks.<ref name=Hogan /> Impedance control has served as the basis for much of the work done in the area of determining how humans interact with their environment. The work of Neville Hogan has been particularly useful in this area as much of the work being done today in this area is based on his previous work.<ref name=Hogan />

== Applications in robotics ==

=== Neuroprosthetics and exoskeletons ===
[[File:2005-11-14 ShadowLeg Finished medium.jpg|thumb|A robotic leg powered by [[Pneumatic artificial muscles|air muscle actuators]] ]]

	Knowledge of human stiffness variation and stiffness selection has influenced [[robot]]ic designs as researchers attempt to design robots that act more like biological systems. In order for robots to act more like biological systems, work is being done to attempt to implement stiffness modulation in robots so that they may interact more effectively with their environment.

	State of the art [[neuroprosthetics]] have attempted to implements stiffness control in their robotic devices. The goal of these devices is to replace the limbs of amputees and allow the new limbs to adjust their stiffness in order to effectively interact with the environment.<ref name= Fite>{{cite journal|last=Fite|first=Kevin|coauthors=Mitchell, J.; Sup, F.;Goldfarb, M.|title=Design and control of an electrically powered knee prosthesis|journal=Rehabilitation Robotics Conference|year=2007}}</ref>

	Additionally, robotic [[Powered exoskeleton|exoskeletons]] have attempted to implement similar adjustable stiffness in their devices.<ref name= VanDerKooij>{{cite journal|last=Van Der Kooij|first=H.|coauthors=Veneman, J.; Ekkelenkamp, R.|title=Design of a compliantly actuated exoskeleton for an impedance controlled gait trainer robot|journal=Engineering in Medicine and Biology Society Conference of the IEEE|year=2006|pmid=17946801}}</ref> These robots implement stiffness control for multiple reasons. The robots must be able to interact efficiently with the external environment but they must also be able to interact safely with their human user.<ref name= Kazerooni>{{cite journal|last=Kazerooni|first=Homayoon|title=The human power amplifier technology at University of California, Berkeley|journal=Journal of Robotics and Autonomous Systems|year=1996}}</ref> Stiffness modulation and impedance control can be leveraged to accomplish both of these goals.

	These devices achieve variable stiffness in various ways. Some devices use [[Controller (control theory)|controllers]] and rigid [[Servomechanism|servomotor]]s to simulate variable stiffness. Other devices utilize specific flexible [[actuator]]s to achieve a various levels of limb stiffness.

=== Actuation Techniques ===

	These robotic devices are able to achieve variable stiffness through various mechanisms such as simulating stiffness variation through control of stiff actuators or by utilizing variable stiffness actuators. Variable stiffness actuators mimic biological organisms by changing their inherent stiffness.<ref name=VanHam /> These variable stiffness acutators are able to control their inherent stiffness in multiple ways. Some vary their stiffness much like humans do, by varying the force contribution of antagonistic mechanical muscles. Other actuators are able to adjust their stiffness by taking advantage of the properties of [[Plasticity (physics)|deformable]] elements housed within the actuators.

	By utilizing these variable stiffness actuation technologies, new robots have been able to more accurately replicate the motions of biological organisms and mimic their energetic efficiencies.

== See also ==
*[[Powered exoskeleton]]
*[[Neuroprosthetics]]
*[[Robotics]]
*[[Stiffness]]
*[[Motor control]]
*[[Neuroscience]]
*[[Motor coordination]]

== References ==
<references/>


[[Category:Neurophysiology]]
[[Category:Limbs]]