Concepts in HMS – Stability
1 Introducti on
Stability is seen as situation; it depends on posture, while in the latter it is seen as a personal
characteristic or capability
2 Stability, a mechanical perspecti ve
Inverted pendulum = a model for a single body segment or several segments over a single joint. It
has a mass (m), a height for a center of mass (h) above the hinge joint (R). A hinge joint only has one
degree of freedom.
Kinematic state of the pendulum = a combination of kinematic variables as orientation (Beta), angular
velocity (db/dt) and agular acceleration (d 2beta/dt2).
There are several states for the pendulum. The first one is in static equilibrium (oriention is constant
and velocity and acceleration is zero). This can only be the case when the pendulum is upright.
Acceleration would occur when the pendulum is moving in a direction and the pendulum would
move in a new orientation in which the gravity would start to exert a moment. So dM/dBeta is
positive and the inverted pendulum is unstable. A system is stable when it can successfully return to
its panned state after a small perturbation.
You can also describe stability in terms of potential energy of the pendulum. A system is stable when
its potential energy is at a minimal and hence increases when the system is perturbated. So the term
‘mgh’ will decrease with any perturbation, since the height of the centre of mass of the pendulum
will decrease. The potential energy is maximal in the upright position and would be minimal for a
pendulum hanging down form hinge joint R.
The moment grows with the perturbation magnitude. A spring exerts a force that is a function of the
change of its length relative to its rest length, but the sign of the force is opposite to that of the
length change. The length of the spring will change with each perturbation, the length change will
cause a force Fs in the spring that will exert a moment on the pendulum. Is the Ms is larger then the
moment exerted by gravity dM/dBeta is negative and the pendulum is stable. In terms of potential
energy, the increase in elastic potential energy due to the perturbation is larger than the decrease in
gravitational potential energy.
Stiffness (K); for low K, the system is unstable and a very large deviation of the orientation occurs,
which ends with the pendulum flat on the floor. For large K, the inverted pendulum is stable, that is
the displacement is bounded and dependent on the magnitude of the perturbation. For higher K, the
displacements of the pendulum are smaller and the frequencies of the oscillations are higher.
Increasing the moment arm (a) of a spring will increase the effect of the spring, and does this
quadratically.
Dampers = elements that produce a force, with a magnitude depending on the rate of change in
length of the element and a sign opposite to that of the velocity of the length change. Dampers
dissipate kinetic energy and transform it to heat. Oscillations will always disappear more or less
gradually due to frictional losses of kinetic energy. Dampers enhance this effect by absorbing more
kinetic energy.
,Performance of a system = the magnitude of resulting displacement and the rate at which the system
returns to the planned state, or the number of oscillations after a perturbation.
Robustness = the property that describes the maximum perturbation that a stable system can
tolerate, in other words; describes to what extent a system will remain to be stable after damage has
occurred.
Lumped effect = the mechanical behavior of a joint is often modeled as a lumped effect of all the
structures spanning the joint. The contribution of each structure spanning the joint is dependent on
its stiffness or damping coefficient and its moment arm. This effect describes the net effect of all
structures around the joint as a combination of a rotational spring and a rotational damper.
3 passive ti ssues in stability
Passive tissues act as springs that can contribute to stability. These tissues also provide damping due
to their visco-elastic behavior. This refers to the fact that the force in the structure depends not only
on its length (elastic) but also the rate of length change (viscous). Laxity is seen as the determinant of
stability. Passive joint stiffness indeed contributes to stability, but is not the only contributor.
Ligaments and joint capsules show non-linear behavior, with low stiffness at low lengths and high
stiffness at high lengths. This is highly functional as it allows joint movement without much resistance
by these structures in most of the range of motion of the joints.
The increase in force in a stretched ligament would provide a moment that is lower then the moment
due to gravity (dMs/dBeta < -dMg/dBeta) and thus the total moment would increase in the direction
of the perturbation (dM/dBeta > 0). The role of passive structures providing stability is limited by the
fact that these tissues show quite substantial creep deformation; when they are continuously loaded,
their length will gradually increase. Consequently, their effectiveness as a spring will gradually
decrease under sustained loading. Injury or degeneration of joint structures may cause problems in
control of joint motion and even instability. Partial or complete ruptures of ligaments will reduce their
stiffness, as well as degenerative changes.
4 muscular contributi on to stability; co-contracti on
For most joints, ligaments do not provide sufficient stiffness to stabilize joint positions or movement
against gravity. Muscle stiffness is the net result of the mechanical properties of the contractile
element and series elastic element of the muscle. The mechanical properties of the contractile
element depend on the number of coupled cross-bridges and hence on muscle activity. The force-
length relationship of muscles can contribute to spring-like behavior of active muscles, as it implies a
change in active force dependent on changes in length. Active muscles also provide some damping
due to frictional losses within the muscle tissue and due to the active force-velocity relationships of
the muscle. Active muscles can be seen as springs and dampers that act in parallel with the
ligaments. The stiffness and damping of muscles can be controlled by the CNS, since they increase
with the level of muscle activity. The passive and muscle stiffness and damping are called intrinsic
stiffness and damping. These result in forces counteracting a perturbation purely based on mechanics
of the tissues and thus the resulting moments counteracting the perturbation occurring immediately
without any delay.
Co-contraction = simultaneous activation of agonists and antagonists, which is often used to stabilize
a joint. For example to keep the trunk in upright position, because in these postures little or no
muscle activity is required to equilibrate the trunk against gravity. However, low muscle activity
implies low muscle stiffness and thus making a unstable situation. So muscle activity needs to be
, increased on both sides of the joint. When mass is added, it increases the moment due to gravity
when the trunk deviates form its upright posture and needs to be compensated by increased co-
contraction to maintain stability. At larger angles of trunk flexion or extension, no co-contraction is
needed, here the muscle force required to provide equilibrium against gravity is high enough to also
provide sufficient stiffness for stability.
Muscles with large moment arts and short lengths have the largest stabilizing potential, but this
depends on the posture. Local muscles connect each segment, global muscles cross several joints.
Activity of the local muscles is prerequisite to stable such a system.
Lack of precision of muscle force is the variability of force. The magnitude of this variability has been
shown to increase with fatigue, as well as with aging and exposure to stressors (fear of secondary
tasks). Force variability increases with increasing target force (single-dependent noise). As a
consequence of this, the amplitude of internally generated perturbations increases with muscle
activity and thus also with co-contraction. Then learning a new precision task co-contraction was
initially high and gradually decreased.
In dynamic situations, creating high joint stiffness may not be possible, as it would only restrict the
movement. Creating a robust system by means of co-contraction is energetically inefficient. Although
a system needs to be stable, it may be more efficient to wait until a perturbation occurs and correct it
by means of reactive control that can then be specific to the magnitude and direction of the
perturbation. So the effect of co-contraction is different for the system trunk than for the system
whole body.
5 muscular contributi on to stability; feedback control
Stiffness and damping provided by feedback mechanisms can be ued in actively controlled systems.
Feedback control consists of active corrections of errors in the state of a system based on
measurements of the state and comparison to the planned state. Feedback control causes forces
counteracting a perturbation at some delay after the perturbation, since transmission and processing
of information in the nervous system takes time.
A feedback loop can be characterized by its gain, the ratio of the magnitude of the correction made to
the magnitude of the error. For positive feedback, the gain is thus the equivalent of stiffness and for
velocity feedback, it is the equivalent of damping. Feedback control always acts at a delay as it takes
