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Knee-inspired Tensegrity Flexural Joint


Abstract and Figures

This paper proposes a prototype tensegrity flexural joint, which has a kinematic behavior inspired by a human knee. The paper presents the predicted and actual motion of several prototype knee designs during knee flexion. Most robotics literature views the human knee as a revolute joint or as a ball and socket joint, which have limited range of rotation. A knee design which considers the hybrid (flexible-rigid) structure of the knee would be able to better approximate real knee behavior and hopefully lead to a better design of artificial (prosthetic) knees.
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Knee-inspired Tensegrity Flexural Joint
Erik Jung1,2, Victoria Ly1,2, Dennis Castro1,2, and Mircea Teodorescu1,2
Abstract This paper proposes a prototype tensegrity flexu-
ral joint, which has a kinematic behavior inspired by a human
knee. The paper presents the predicted and actual motion
of several prototype knee designs during knee flexion. Most
robotics literature views the human knee as a revolute joint
or as a ball and socket joint, which have limited range of
rotation. A knee design which considers the hybrid (flexible-
rigid) structure of the knee would be able to better approximate
real knee behavior and hopefully lead to a better design of
artificial (prosthetic) knees.
The human leg has a couple primary joints that facilitate
gait. [1] The hip joint is crucial to achieve a wide range
of motion, and assists with walking. The knee plays a
crucial role by helping reduce the impact of body weight
on the feet while walking. Each type of tissue is unique
due to its structural and material properties. Because of this,
anatomical joints allow the body to sustain most impacts.
The current design proposes a multi-axis structurally com-
pliant joint (Figure 1) which provides protection through
imposed movement. One of the major advantages of tenseg-
rity inspired robots is that the stiff compression elements
support the load, the elastic tension elements deform and
distribute the load throughout the entire structure absorbing
impacts like its biological counterparts. [2][3] This paper
represents our first steps towards designing a biologically-
inspired tensegrity flexural robotic joint that resembles the
knee joint. This particular design is a learning tool to help
us understand and mimic a human knee hence a hinged
approach. In the future, we hope to build a self-standing
actuated model.
Tensegrity (short for ”tensile with integrity”) inspired
robots are inherently flexible structure consisting of a series
of compression elements suspended in a network of tension
elements. As a result the applied forces or impacts are
distributed throughout the structure via multiple load paths
and impacts. [4][5] The pairing scheme of our tensegrity joint
is inspired by the muscular and fascial connections within the
human leg. Since all actuation is cable driven, the proposed
design has the motors located off of the robot itself which
reduces the weight of the robotic structure.
1University of California, Santa Cruz, Santa Cruz, CA 95064, USA,,,
2Dynamics Autonomous Navigation Surface Engineering and Robotics
(DANSER) Lab at University of California Santa Cruz
Fig. 1. Proposed tensegrity knee joint, connecting femur and tibia
equivalents. The joint is controlled by an active pair of strings tuned for
stiffness creating a variable level of flexibility within the structure.
A. Compression Elements
The proposed design consists of two compression el-
ements acts like the bone structure in human legs. The
upper compression element is equivalent to the femur bone,
while the lower combines the tibia and fibula into one
single component. The tensegrity structure anchors the femur
compression element while also routing the tension elements
that suspend the lower half of the leg to produce flexion.
B. Tension Elements
The tension elements use cables (fish-line spectra cord
and bungee cord in our physical model), and belong to
two categories: active and passive. The active strings are
coupled into antagonistic pairs, allowing the structure to cre-
ate motion similar to leg muscles. Passive strings resembles
the biological fascial connections of the the knee joint: the
tendons and ligaments. The structure elastically deforms in
response to the actuation of the active muscles, and conform
the leg back to equilibrium.
C. Design and Control Through Simulations
Fig. 2. The first iteration, A, was a simple model of the leg only capable of
standing in equilibrium. The next model, B, was the second design iteration
which more accurately resembled the muscles in the leg. The third model,
C, was geared towards a more biologically accurate muscle system. The
last model, D, was capable of knee flexion with the addition of biological
1) NASA’s Tensegrity Robotics Toolkit (NTRT): We sim-
ulate system behavior using NTRT (Figure 2) to simulate
contacts and impacts between elastic and rigid body dy-
namics. [6][7] This simulation environment uses a Cartesian
mapping system to describe the geometrical shape of a
tensegrity structure, Euler-Lagrange formulation to describe
the dynamics to predict the elastic forces developed inside
elastic cables. [8][9]
f=(k(xli) + b˙x x > li
x=||pi,0pi,1|| (2)
where kmeasures the spring stiffness, xrepresents the
Euclidean distance between the spring-cable assembly, bis
a term for linear damping, and liis the cable length.
From the first iteration (Figure 2A) the models need a
rigid placement to the environment since the change in center
of gravity causes the model to fall over. Figure 2D, we
considered the model being attached to the upper most point.
This achieved the lower leg displacing towards the upper leg,
and the tension elements required closely represent the bicep
femoral muscles contracting. From the final iteration seen in
Figure 2D the hung model was developed into two different
joint revisions where the first, REV 1, and in the second
revision, REV 2.
After testing different designs in the NTRT environment,
we found that the last model (Figure 2D) has suspension
that achieves flexion with the bicep femoral muscle group.
This aligns with the anatomically correct muscles required
for flexion. For prototyping, we constructed our model using
dowel sticks and 3D-printed joints (Figure 3) to create the
overall geometry. In the first iteration of the physical model
the attachment points at the knee joint became too congested,
which could cause cable interactions. To resolve this issue,
Fig. 3. Close up of the 3D printed end pieces with holes to keep the elastic
cord in the proper line of action.
new end parts (Figure 3) were printed for knee model REV
2 that contained holes to tread the elastic cord through and
still keep it in the correct line of action.
For the actuation a simple torque analysis was done on the
model with a total weight, estimated at 0.74N, with length
of the model at 116 cm. To find max torque needed to lift
the entire model the total weight and length were used in the
following equations:
Equation 4 shows τis Torque, F is the force in Newtons and
L is the length in centimeters. To calculate torque required
by motors: (1) the minimum torque is when the length of the
cable is 0.58mand the total mass is at the end of the length,
(2) The max torque at max length of 1.16mand the total
mass is at the end of the length. The peak torque required
is a maximum of 0.85N·m.
For the NTRT simulations (referring to Figure 2), the
density, stiffness, and pretension reflect the proposed physical
models. We track the coordinate displacement of the end
effector on the base of the leg in the last two model revisions.
Referring to Figure 4, there are significant differences
between the range and steadiness of motion. The plots
show convincing evidence that the degree positioning of
where the compression elements are angled provide the
tensegrity structure with a wider range of motion. The motion
shows REV 2 with smooth parabolic displacement during
the flexion with actuated muscles, while the REV 1 lacked
that particular feature in its results. Each tensegrity structure
performs its task of flexion through the simulated actuation
of cables to act as muscles. Using the NTRT simulation, we
built a physical model to simulate a working tensegrity knee.
To better analyze the positional displacement of the phys-
ical model we used a motion tracking software, OptiTrack.
With this data we calculated the actual angle from start to
finish of our first and second physical prototype. Our first
design was only able to achieve a maximum angle of 91
degrees, whereas with the new design we achieved an angle
of 129, which is not the complete 160that the human
knee can flex, but does show similar correlations existing
knee prostheses which achieve 120[10]. Referring to 6,
REV2 reaches a higher point in the y-axis while the femur
maintains a position closer to equilibrium. REV1 happens to
not maintain the initial equilibrium on the femur and stresses
the compression elements. This specific tensegrity leg was
Fig. 4. NTRT simulations, of the node located at the base of the tibia
tensegrity leg model (see Figure 1). Under the effects of gravity and
applied the same controller for both models, we can see that REV2 reaches
significantly further than REV1.
Fig. 5. In this figure, we tracked the displacement of the knee as it flexes.
The femur in REV2 does not move as much as REV1, which meaning our
revised structure is more stable. REV 2 is able to achieve a larger range
of motion than REV1, and stays on its motion path throughout the whole
flexion of the tensegrity knee.
created to maintain structural stability while being able to
flex and put stress on the passive tensile elements on the
structure. The passive tension components allow the structure
to absorb external forces and distribute it evenly across the
knee joint.
The process towards calculating the angle displacement
shown was by using the Tracking Marker: Femur 4 in Figure
1 as a point of origin. In both mechanical revisions, for the
tensegrity structure, we used the same location for the mark-
ers and calculated the angle based off of the displacement
from the Tracking Marker: Tibia 3 in Figure 1.
cos(α) = ¯x·¯y
We found the angle of displacement for both REV1 and
REV2 from the motion capture. For the physical models,
we also tracked the ”bend” of the upper leg segment that
occurred when the flexion was at its peak, and the angle of
flexion relative to initial position of the upper leg segment.
We are able to see a significant difference between the
equilibrium and flexed position from Figure 6.
The current research proposed a prototype tensegrity flexu-
ral joint, which has human knee inspired kinematic behavior.
Using NTRT, as a real-time simulator, we were able to test
Fig. 6. Using the OptiTrack motion system, we aligned the initial position
of both REV1 and REV2 of the tensegrity leg. This flexion period of both
models show a significant difference in the final position.
the motion and efficiency of our tensegrity designs under
the influence of gravity and external forces. Through the use
of OptiTrack, we were able to get precise tracking on the
displacements and difference between each version of our
structure. In the future, we plan to take what we have learned
from the tensegrity knee and apply it to hip and ankle joints
as it would be more useful if we can analyze a completed
tensegrity leg.
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... Tensegrity ("tensile with integrity") [8] is considered to be a solution to the combination of hard and soft components. Tensegrity robotics ranges from robots created for exploration through rolling locomotion [9], [10], [11], to bioinspired structures [12] and robotic systems that resemble different joints in the human skeleton [8], [13], [14], [15], [16]. This particular type of robotics consists of rigid and elastic elements which work in unison to return to equilibrium when acted upon by an outside force. ...
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