Tensegrity robots : Assur tensegrity structures mimic the motion of a caterpillar

Butterfly caterpillar Monarch Donaus Plexippus © 2010 Tom Murray

One of the very interesting aspects of learning about tensegrity is how multidimensional the process becomes. Kenneth Snelson coined the phrase, floating tension to describe his early sculptures when he first started playing with tension compression models in 1948. Little did Snelson realize that his shape tension compression discovery would lead to such a diversity of architectural and relevant descriptions about shape sensing performing within Nature. In recognition to his fundamental discovery I introduce the term, tensio natat, floating tension. It’s way more than just about shape, it’s how nature uses shape as a way to solve problems, like a small arthropod caterpillar crawling , employing the principals of tension integrity, crawling over the surface of a leaf. How can such a soft creature with no hard skeleton move with such liquid smoothness, what is the nervous system that controls such slinky motion?

As an observer of Nature if you really want to demonstrate something as simple as a caterpillar crawling then try building a robot that accomplishes just such a locomotion, just try. If you were a NASA supervisor describing a remote terrain vehicle your description profile would be very close to what a caterpillar accomplishes. The imaginary engineering mandate for prospective robot builders would highlight, the potential of the robot to perform …”astonishingly efficient gait and capable of maneuvering mobility over obstacle strewn, rough terrain.” Such a robot has been built in Israel I will be describing from the Bioinspiration & Biomimetics. 7 (2012) 04006 (9pp) authored by O Orki, A Ayali, O Shai and U Ben-Hanan in Modeling of caterpillar crawl using novel tensegrity structures.

Bamboo tensegrity robot “Shi” Yogyakarta, Indonesia, August 2012

Engineers are constantly fascinated with how insects move, first with multi legged motion and now with the slithering efficiency of caterpillars. From their engineering perspective the real study of caterpillars centers around the generation and control of locomotion within the body of the caterpillar which are soft bodies without any support from a hard skeleton. In order to overcome this pivoting hurdle the caterpillar uses inner fluid and tissue pressure to stiffen the body, permitting muscles to perform work. ” As a result, the contraction of any one muscle affects all the rest, either by altering their length or their tension, which presents major challenges to the control system,” according to the authors. As a caterpillar crawls, viewed from the side a wave of muscular contractions starts at the back end to progress forward to the front. “Anatomically, crawling is achieved by muscles (contracting which are attached to invaginations on the inside surface of a soft and flexible body wall, ” pulling each side tergal segment drawing closer to its nearest neigbour’s tergal segment, then onto the next pair of tergal segments. The simple nervous system controls, “The basic timing and patterning of the rythmic motor pattern is assumed to be generated and controlled by a central pattern generating network (CPG net). Other engineering attempts were previously made using rigid joint elements employing linear actuators by Wang and Stulce. Trimmer developed a deformable caterpillar model but introduced ,’…. considerable control complexities.Yet each of these investigators were not matching Nature’s relatively simpler caterpillars control system and maneuverability.

” The principles of tensegrity can be found at essentially every scale in Nature. At the macroscopic level, the skeleton of vertebrates is compressed and stabilized by the pull of of tensile muscles, tendons and ligaments. At the cellular level, the cytoskeleton provides a good example: contractile microfilaments provide tensile forces while microtubules provide the opposing compression. At the lower end of the scale, proteins and other key molecules in the body also stabilize themselves through the principles of tensegrity. Tensegrity systems gain mechanical stability by maintaining a pre-equilibrated state using two types of elements: elements that are always tensioned (cables) and elements that are always compressed (struts). This pre-equilibrated state, in which the internal forces balancing compression and tension stabilize the entire structure, is termed pre-stress.”

So far in the cerebrovortex.com essays mostly shape has been discussed  within a static concept but remember the most intriguing thing about Nature, everything is moving. The authors took a similar tact to employ tensegrity as the basis for the robotic motion for controlling a caterpillars segmenting wave shifts slinking along its soft body. ” Movement is achieved by changing the lengths of some of the tensegrity elements, which in tern causes the shape of the robot to be altered. One of the major challenges in the design of such robots is the maintenance of pre-stress forces during motion, which is necessary to preserve structural stability.” Some other robot investigators have tried to resolve this by building actuators to always increase the pre-stress forces only on the cables thereby guaranteeing the robots stability but constraining the robots shape change. Other investigators used minor shape changes which were not far removed from the stability balance point of the structure but sacrificing equilibrium at the same time of the structure if too much motion was present. The authors opted to mimic Nature to produce their system to have well constrained data, not over constrained and redundant or under constrained and inadequate. Just such a system is potentially possible if only force equilibrium is achieved around each tensegrity joint, ‘to create a well-constrained system termed a statically determinate structure. ‘ The authors chose to work around a novel statically determinate tensegrity structure, by addressing a special type of determinate structure called an Assur structure which exhibits a specific property of shape, in which, a special configuration, termed a singularity in which pre-stress is present in all the elements and if the three component legs meet at a single point.

(a) Triad general configuration (b) Triad in singular configuration (c) Assur tensegrity structure triad truss meeting at one point in space

Assur structures exhibit specific properties, …”with the removal of any rod results in a mechanism composed of all the other rods (those remaining rods involved with the segments).” The triad truss triangle accomplishes this by not leaving out any rods which might be isolated as immobile, outside of the tension net sustained within all the remaining rods, 5 in this case out of 6. In other words, changing the length of any rod will result in motion of all remaining rods. In 2010 it was proved that every Assur structure can assume a special configuration (called a singularity) in which pre-stress is present in all elements. Singularity is obtained when the continuation of the three ground legs intersect at a single point as in figures (b) and (c). Figure (c) is comprised of the tensioned rods, replaced with cables, marked with dashed lines and compressed rods are replaced with struts., compared to (b) which is only composed of all rods. This singular triad configuration structure (c) can sustain pre-stress forces (everywhere, simultaneously in all components, thus both local and distant pre-stress are in harmony) and is termed a Assur tensegrity structure.

The real caterpillar has hundreds of muscles to control within its abdominal linked body segments, each segment involves approximately 70 discrete muscles. ” The major abdominal muscles in each segment are the ventral longitudinal muscle (VL1) and the dorsal longitudinal muscle (DL1). Each segment has its own distinct VL1 and DL1 muscle and each is controlled separately. The VL1 and DL1  attach at the inner stiff ridges on the upper and lower surface of each segment termed tergal and sternal antecosta respectively. The caterpillar model robot was modified to fuse the triad into a single rigid un-actuated bar and cables attached at each side to a merged 3 ground supports like a T on its side rotated 90 degrees, with legs connected at the bottom of each T bar.

Caterpillar muscles bounded by tergal and sternal antecosta per segment with Assur tensegrity structure transformed into Bar connected  with cables to Strut segments, all pre-stressed

The structure of the caterpillar robot is mimicked in each segment, “The upper cable assumes the role of the DL1 and the lower cable represents VL1 longitudinal muscle. The strut, which is always subjected to compression forces, represents the hydrostatic skeleton, and segment legs are for support and grip similar to the biological caterpillar prolegs.”

Caterpillar robot Assur tensegrity control algorithm

I think it is important to dwell on the actual  inspiration of the tensegrity control as a coordinated sequence dependent on the altering shape of the segments as they squeeze into a short space and the net result is that the robot glides along a planar surface. How these researchers first separated the control pacing into both local control coordinated with central control reveals a lot of how parts of how a small central nervous system functions. Also mimicing the caterpillar they divided the control schema into two levels: “high level control and low-level control.”

“Low-level control is inspired by the mechanical characteristics of the caterpillar. It is composed of localized controllers for each of the strut, cable and leg elements.
Each controller is independent of all the others: the controller output of an element is calculated using only the inputs of that specific element. The strut controllers simulate the internal pressure of the hydrostatic skeleton, the cable controllers simulate the elastic behavior of the muscles and the leg controllers simulate leg behavior. High-level control simulates the function of the nervous system. The role of the high-level control unit is to deliver commands to the cable and leg controllers in order to coordinate motion. Just as the internal pressure of the caterpillar is not directly controlled by the nervous system, struts are not driven by high-level control. High-level control is also divided into two levels: levels 1 and 2. Level 1 is the central control unit and is inspired by the caterpillar’s central pattern generator (CPG) for locomotion. Its role is to control the timing of movements and to activate the relevant cables and legs. Level 2 control mediates between level 1 control and the cable controllers of each segment, adjusting level 1 control for each segment according to terrain. Segment control is thus local as inspired by caterpillar segmental ganglia. An essential property of high-level control is that the coordination of locomotion is triggered by the contact of the legs with the ground. Figure 6 summarizes the control hierarchy.”

Perhaps a reader of this essay is now asking why so much detail of the control sequence can’t it be summarized more briefly? Here’s my point. These authors have singled out in their solutions the critical division of pacing with sensing during the locomotion sequence. It’s precisely this  sort of thing that Nature did a long time ago to derive the details of moving a tensegrity creature. Through robots mimicking a caterpillar moving we learn the hierarchical command structure that separates local control from central control on a shape changing machine. I find their solutions fascinating that these authors keep referring to the caterpillars arsenal of control pacing network for inspiration to solve moving the robot.

“In general, robot degrees of freedom (DOFs) can be controlled by one of two control types: motion control or force control. In motion control the controlled variables are kinematic (position, velocity and acceleration); in force control the controlled variable is the force the robot exerts on the environment. Motion control is useful for many industrial applications because of its high accuracy: each joint position is calculated and monitored at each point in time . Nevertheless, this type of control is not well fitted to the nature of soft robotics. Soft robots deform by external and internal forces, which makes it very difficult to control the exact motion parameters of the robot’s DOFs at each point in time. The more suitable type of control for soft robots is force control. In our model, cables and struts employ a force control scheme based on impedance control. The general control law for the low-level controllers is

Fout = F0 + k(l − l0) − bv

The output force Fout is a sum of three terms: F0 is a constant and initial force which has the role of maintaining the pre-stress forces inside the tensegrity segments. F0 is negative for cables (tension forces) and positive for struts (compression forces). The term k(l−l0) is the static (or elastic) relationship between the output force and length, also known as stiffness. This term causes spring-like behavior:when the element length increases, the output force is also increased and vice versa. The degree of stiffness is controlled by changing the stiffness coefficient (k). Finally, bv is the relationship between the output force and velocity. It functions as a damper in order to avoid fluctuations and to moderate element reaction time. (It may also be thought of in terms of viscosity.)

“Cables simulate the function of caterpillar muscles. Biological caterpillar muscles have a large, nonlinear, deformation range and display viscoelastic behavior. If all controller parameters (F0, k, l0 and b) remain constant, the model will remain in steady state and will not move. Cable behavior is controlled by input signals from highlevel control to cable controllers (which we call a high-level command).
As high-level control simulates the nervous system and cables simulate muscles, the high-level command simulates nerve stimulation. The ‘command’ input receives values between 0 and 1. A command value of 0 indicates that the cable should be ‘relaxed’ (i.e. a low value of k and a high value of l0). A command value of 1 indicates that the cable should be ‘tightened’ (i.e. a high value of k and a low value of l0). Intermediate values indicate intermediate behavior. A low-pass (LP) filter is placed between the command and the cable controller. This LP filter slows down cable reaction, which simulates the slow reaction of the caterpillar muscle.”

“Struts have simpler behavior than cables, struts simulate the internal
pressure of the hydrostatic (caterpillar’s) skeleton. In the biological caterpillar, the internal pressure is not isobarometric and the fluid pressure changes do not correlate well with movement. For simplicity, our model assumes nearly constant  pressure. The stiffness coefficient (k) is set to zero, and the  control law for struts is reduced to

Fout = F0−bv

with positive values of F0 (compression force). Strut parameters stay constant during locomotion and are not driven by high level control. Legs are not part of the tensegrity triad and are not impedance controlled. In our model, legs have only two positions: lifted and lowered. The transition between these positions is controlled by a simple motion controller. When a leg touches the ground it is ‘planted’ and cannot be lifted until the next stride. This behavior is, again, modeled after the biological caterpillar.”

” In order for effective locomotion to occur, the model’s stride is divided into three phases. In the first phase, the three posterior segments of the caterpillar are lifted
and shrunken. This phase ends when the most posterior leg is lowered and touches the ground. In the second phase, the crawling wave passes through the body. In the third and final phase, the three anterior segments are lifted and expanded one after the other. The simulation is programmed such that the next stride begins before the previous one is completed.

When crawling, the segment length changes by an average of 31%. This result is consistent with the observation of caterpillar muscles, which exhibit comparable shortening to 30% of the resting length. The duration of one model stride is also very close to that of the biological caterpillar— 2.71 and 2.78 s, respectively (a difference of 2.5%).”

The authors summarized their control findings with, “We found that impedance control keeps the Assur tensegrity structure in a singular configuration, thus maintaining the stability of the structure. In addition, impedance control enables us to produce a soft model with a controllable degree of softness. While impedance control is complex in typical industrial robotics, it is much simpler to implement in tensegrity structures. For reasons outlined below, impedance control is a ‘natural’ choice for our model.

1. In tensegrity structures, each element is controlled separately and independently. This is in contrast to a standard industrial robot in which all DOFs are conjugated.
2. There is no need to transform end-point forces to actuator forces, and there is no need to consider robot dynamics in the impedance control law; the equation is used as is.
3. All tensegrity structures have infinitesimal motion in their singular configuration. The actual motion around this point is determined, among other things, by the elasticity of the materials (cables and struts). Impedance control can be thought of as a way to increasing this elasticity (where the stiffness coefficient k is equivalent to Young’s modulus).”

“The model exhibits several characteristics which are analogous to those of the biological caterpillar, empirical testing of the model has demonstrated that effective crawling requires that each stride be executed in three different phases. Trimmer et al examined the kinematics of the biological caterpillar and found kinematic differences between three anatomic parts of the caterpillar: the thoracic segments, the midbody segments and the posterior segments. This distinction is similar to the three stride phases of the caterpillar model. The caterpillar model can navigate different terrains and in different directions using the same crawling pattern without adjusting the control scheme. This is made possible by slow stride speed and firm ground planting. The same is true for the biological caterpillar.
The internal pressure of the biological caterpillar is not a function of its size. During growth, its body mass is increased 10 000-fold, while internal pressure remains constant. In the same way, our model is able to use the same pre-stress forces regardless of the model size (although the pre-stress forces must excide a certain force threshold).
It was proposed above that the mechanical properties of caterpillar muscles may assume responsibility for some of the control tasks otherwise carried out by the nervous system. Our model demonstrates that using impedance control for each caterpillar muscle force development is about four to seven times slower than that of an insect flight muscle. The model shows that adding the low-pass filter to the cable controller, which makes the cable react slower, eases highlevel control and results in smoother motion. Note that the time constant of the filter was determined empirically in order to optimize results. Only afterward was the comparison made to the biological muscle,  both of which exhibited similar time constants. In addition, other crawling parameters related to time (e.g., the duration of one stride) are comparable in both the model and the biological caterpillar.”

“In our model, only three segments are contracted.The reason for this limitation is that, when four segments (like a real caterpillar) are contracted and lifted together, the impact of gravity becomes much larger (especially in phases 1 and 3). This makes it difficult to program locomotion in a way that will be robust in all crawling directions (vertical and upside-down). This discrepancy should be improved in future versions of the model.

In summary, our model is consistent with many of the actual biomechanical attributes of the Manduca sexta caterpillar. Our research further suggests a few characteristics that the biological caterpillar may possess. In the model, stride timing is strongly dependent on the signals that the legs send when touching the ground. Without those signals, locomotion is not robust—it tends to be inefficient and many times unstable. Observations show that feedback from the legs is not essential for maintaining locomotion gait in fast insects like the cockroach, while it is critical in slow insects like the phasmid (the stick insect).

Although we were not able to find similar information on caterpillars, their slower gait makes it reasonable to assume that they also need leg feedback. The model introduced in this paper strongly supports this hypothesis.”

In other words the robot caterpillar needs either a primitive proprioceptive mimicking system to provide leg feedback plus a primitive vestibular system could also be built in, to also stabilize the position of the entire body while  the caterpillar robot segments slink across a surface. The basis of their design is centered on a Assur tensegrity structure that is in pre-stress balanced through out the entire structure all segments attached yet locally and independently controlled in their execution  of the locomotion sequence at the same time central pattern generators as Level 1 Central Control are initiating before each phase segment motion is completed. The robot caterpillar works is based on a very mechanical sensing of tensegrity elements. Small control systems mimic simple nervous systems based on tensegrity incorporation into the designs employ shape sensing  as changed pre-stress elasticity.

So what does this teach us about concussions? Our brains are massive tensegrity structures built on eons of empirical designed tensegrity singularities controlled both locally and centrally within pre-stressed networks that sense shape changes at the protein dimension within brain networks and execute the output at the macro level like vision and vestibular/otolith reflexes that go off because the concussion has perturbed the tensegrity Level 1 control network, see the connections that robot caterpillar Assur tensegrity control networks teach us?

As a final description let me quote from Dimitrije Stamenovic and Donald  Inber’s Tensegrity-guided self assembly: from molecules to living cells from Soft Matter, 2009, 5, 1137-1145. Here is a particularly pertinent observation from their paper.

“Because molecular assembly events are influenced by force and mechanical loads are distributed in specific patterns across cytoskeletal elements, biochemical reactions can be induced to proceed in specific patterns that precisely match the needs of the cell to resist those applied stresses. Due to the existence of a complementary force balance between microtubules, contractile microfilaments, intermediate filaments and membrane adhesion complexes, forces can be shifted back and forth between complementary load-bearing elements and thereby alter their orientation and self assembly. This simple mechanical balance has important physiological relevance for cells, tissues, and organs. One simple example is how transferring forces off microtubules and onto ECM (extra cellular matrix) adhesions decompresses microtubules and thereby promotes their assembly and nerve growth. In fact, nerves will extend new processes in whatever direction tension is applied to their surfaces through ECM adhesions, and nerve fiber patterns map out minimal paths that correspond to tension field lines in whole organs, including the brain, which appear to be driven by mechanical energy minimization.
Another example is the observation that internal microtubules become compressed and buckle when beating heart cells contract, and they undergo elastic recoil and help the cell restore its extended shape when they relax. Moreover, abnormal increases in self assembly of microtubules in heart cells impairs contraction (due to increased internal resistance) and leads to heart failure in whole animals.”

Did you notice the term tension field lines in whole organs like brains? What we have is engineers designing a caterpillar robot using tensegrity parameters to accomplish shape change motion controlled to mimic Nature’s design in terms of pacing coordination. Then there is their reference to beating heart cells undergoing elastic recoil allowing the heart cells to restore their shape when they relax, which is another characteristic within a tensio natat network. Like I said engineers mimicking a caterpillar robot reveals a lot about how our brain and heart fit into a tensegrity framework. If you want to understand concussions at their basic mechanisms, first you have to understand the mechanism itself as operating within a tensegrity pre stressed framework, all elements sensing tension. The engineers are showing us how to build such mechanical robots but the real thing is so much more marvelous and mesmerizing because it builds itself.


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