Yildirim Hurmuzlu, Systems Lab, Southern Methodist University, Dallas, Texas,
USA.

# Magnetically Actuated Millimeter-Scale Biped

Adam Cox11affiliationmark:  Sinan Beskok11affiliationmark:  and Yildirim
Hurmuzlu11affiliationmark:  11affiliationmark:  Southern Methodist University,
USA<EMAIL_ADDRESS>
###### Abstract

This paper introduces a new approach to studying bipedal locomotion. The
approach is based on magnetically actuated miniature robots. Building
prototypes of bipedal locomotion machines has been very costly and overly
complicated. We demonstrate that a magnetically actuated 0.3 gm robot, we call
Big Foot, can be used to test fundamental ideas without necessitating very
complex and expensive bipedal machines. We explore analytically and
experimentally two age old questions in bipedal locomotion: 1. Can such robots
be driven with pure hip actuation. 2. Is it better to use continuous or
impulsive actuation schemes.

First, a numerical model has been developed in order to study the dynamics and
stability of a magnetically actuated miniature robot. We particularly focus on
stability and performance metrics. Then, these results are tested using Big
Foot. Pure hip actuation has been successful in generating gait on uphill
surfaces. In addition, complex tasks such as following prescribed gait
trajectories and navigating through a maze has been successfully performed by
the experimental prototype. The nature and timing of hip torques are also
studied. Two actuation schemes are used: Heel Strike Actuation and Constant
Pulse Wave Actuation. With each scheme, we also vary the time duration of the
applied magnetic field. Heel Strike actuation is found to have superior
stability, more uniform gait generation, and faster locomotion than the
Constant Pulse Wave option. But, Constant Pulse Wave achieves locomotion on
steeper slopes.

###### keywords:

Bipedal Locomotion, Magnetic Actuation, Impulsive Actuation, Millirobot, Gait
Dynamics, Impact Mechanics

##  1 Introduction

The earliest studies of bipedal walkers began with inverted pendulum models.
As research in bipedal locomotion advanced, investigators faced difficulties
due to prohibitively large energy demands and very high component costs when
attempting to develop experimental prototypes. Complex numerical models of
bipedal walkers have been created to better understand the dynamics behind
such systems. Major concerns of modelling have been the model discontinuities
due to feet impact with the walking surface and stability analyses of these
highly nonlinear systems. The main idea of the present paper is to answer key
questions regarding stability and actuation of a very simple bipedal walker.
In addition, we develop a magnetically actuated miniature robot as an
experimental platform to produce a low cost, 3D printed prototype. Using this
platform, we apply the results of the theoretical analyses to a real system
and demonstrate its physical utility. To the best of our knowledge such an
approach to verifying theoretical results using a magnetically actuated
miniature bipedal prototype has never been taken before.

Robotic bipedal locomotion literature is vast, it is impossible to refer to
all work performed on this subject in the confined space of the present
article. We refer readers to books and survey articles written on the subject
and references within (Hurmuzlu et al. (2004); Yamamoto et al. (2020); Grizzle
et al. (2014); Tazaki and Murooka (2020); Mikolajczyk et al. (2022); Hobbelen
and Wisse (2007); Brogliato (2018)). The past research has been mainly
concerned with the study of topics of stability, nonlinear dynamics, and
control of robotic bipedal locomotion. In terms of stability, defining
periodic gait as a limit cycle and analyzing its stability using Poincaré
sections, Floquet Multipliers, and Lyapunov Exponents has become the standard
approach in the field (Hürmüzlü and Moskowitz (1986); Hurmuzlu and Moskowitz
(1987b); Hurmuzlu and Basdogan (1994); Hurmuzlu et al. (1996); Garcia et al.
(1998); Goswami et al. (1997); Ahan et al. (2021); Bruijn et al. (2010);
Ekizos et al. (2018); Saeed et al. (2018); Ramezani et al. (2013); Dingwell et
al. (2000)).

Devising an actuation scheme for a bipedal robot is not a straightforward
task. One of the obvious approaches is to place electric/hydraulic motors and
sensors at the joints. Yet, this renders the robots excessively heavy and
significantly complicates the design. In addition, it results in very high
energy requirements. Passive locomotion (McGeer (1990)), inspired by earlier
works in ballistic walking (Mochon and McMahon (1980)) has emerged as a
promising idea to simplify the design and reduce energy demand. These
mechanisms produce gait without joint actuation. They can walk down an
inclined plane utilizing gravitational energy. Yet, these bipeds have
extremely limited applicability, since they can only walk down inclined
planes. In addition, it is very well known that they have extremely limited
stability range. Nevertheless, researchers have used the basic concept of
passive walking in order to generate active walkers (Tavakoli and Hurmuzlu
(2009, 2010, 2013); Goswami et al. (1997); Garcia et al. (1998); Spong and
Bullo (2005)). Despite these efforts, the presently known walking robots are
extremely complex and have very high price tags (Honda’s Asimo (Honda (2023)),
Boston Dynamics’s Atlas (Dynamics (2023)), Biped Robotics Lab at Michigan’s
Cassie Blue (Grizzle (2021)), Engineered Arts’ Ameca (Arts Limited (2022)),
and NASA’s R5 (Hall (2015))). Consequently, it has become very difficult for
the majority of the researchers in the field to experiment with physical
prototypes.

3-D printed miniature robots can offer an attractive option to develop and
experiment with low cost bipeds using rapid prototyping methods. One can
experimentally study the dynamics and stability of these systems, and develop
control methods and strategies to regulate their motion. Miniature walking
robots have been produced before using soft and flexible actuators. Ijaz et
al. (2020) developed a simple walking robot that uses embedded magnets to flex
the robot’s abdomen to produce locomotion. Baisch et al. (Baisch et al. (2014,
2011); Baisch and Wood (2011); Baisch et al. (2010)) created a family of
walking robots called the Harvard Ambulatory MicroRobot (HAMR). These
microrobots use custom machined piezoelectric boards that flex with an applied
voltage. HAMR was inspired by cockroaches. Miniaturizing actuators for rigid
body robots is difficult. Parker (1995) provide ideas for miniaturizing
actuators. But, for actuators that perform like electric motors, often these
systems use compliant surfaces to allow motion (Buzzin et al. (2022); Fettweis
et al. (2021)). The full extent of these ”muscle-like” walking robots can be
seen in the LCE-microrobots from Zeng et al. (2018). These microrobots are
basically a single torsional muscle, whose design has no other control or
means of locomotion. With rigid actuation, Islam et al. (2022) developed a
minimally actuated, rigid bipedal walker, including five 3D printed rigid
bodies and a single actuator per leg. Each actuator is controlled via an open-
loop sinusoidal profile, thereby eliminating the need for feedback. Their
reasonably small robot (15 cm) was able to walk on level surfaces and produce
turns.

Magnetic actuation of miniature robots has many useful applications in robotic
surgery, drug delivery, small scale manufacturing, and other related
engineering topics (Pawashe et al. (2009)). Use of electromagnetic fields as a
form of robotic actuation allows untethered, low-power mechanisms to regulate
the motion of robots. Yesin et al. (2006) used a combination of Helmholtz and
Maxwell coils to control cubed microrobots in one-dimension, followed by Choi
et al. (2009) in two-dimensions and eventually Jeong et al. (2010) in three-
dimensions. Kim et al. (2013) reduced the number of coils needed for two-
dimensional control by using a pair of coils similar to Helmholtz coils that
could receive independently controlled current. In addition, Mahoney et al.
(2012) explored various magnetic methods of microrobots with helical screw
propulsion and rolling locomotion. Li et al. (2020) created a completely
rigid, two-legged magnetic microrobot that uses periodic magnetic fields to
induce pivot walking to indirectly manipulate cell aggregates, though it is
limited to this one mode of locomotion. While not bipedal, Al Khatib et al.
(2020) developed a rectangular millirobot capable of various modes of
locomotion, including pivot-walking, tapping, and tumbling, with high
maneuverability in tight spaces by capitalizing purely on the torque created
by the magnetic field. They also presented ”stag beetle” and ”carbot” designs
for improved applicability. A special novelty of magnetic actuation is that
these robots are powered externally, but also untethered. This removes the
need for power storage (batteries, compressed air tanks, hydraulic and
pneumatic lines, and any other traditional power source.) or electronics
(sensors and controllers). This allows for novel miniaturization. All that is
required to power the robots on the body itself is a magnet. This opens the
door for rapid prototyping as simpler designs allow better analysis means.

In this paper we present the first magnetically actuated bipedal walking
robot. To the best of our knowledge, this robot is also the world’s smallest
bipedal robot. The robot weighs 0.3g and is 9mm tall. We call this robot the
Big Foot.

The actuation of Big Foot is accomplished by simply placing a set of magnets
in the upper body of the standard passive walker. Next, a magnetic field
applies a moment on the foot-lifting-dynamics to generate gait. The prototype
can be produced within hours using 3D printing techniques at a cost of few
Dollars.

The manuscript begins with a description of the robot design. Next, the
dynamics are presented, including the impact mechanics. Then, the experimental
setup is explained. Finally, the numerical and experimental analyses are
presented.

##  2 Robot Design

Figure 1: Isometric view of the multi-body diagram of Big Foot

The central features of Big Foot are the feet, the Shaft Body, and the
magnets. Actuation is achieved by placing magnets in the Shaft Body shown in
Fig 1. A magnetic field is pulsed around Big Foot to create locomotion. The
Shaft Body’s cylinder is shown in Fig. 2. This cylinder contains outer holes
that magnets are press fit into and a central hole that a low friction shaft
is pressed into. Coincident to the cylinder are legs. These legs have holes
that have a very loose slip fit onto the shaft. There are also collars pressed
on the outside of the shaft assembly to prevent disassembly during operation.
The cylinder, the legs, and the collars are all made from a very low friction
material to allow motion. Finally, at the end of the legs are feet. The
purpose of the feet is to enhance stability. Due to the lack of available
feedback for such a small walker, stability could only be achieved through
physical design. More complicated designs may be considered in future work,
but for this version of Big Foot, the curvature of the feet was defined as a
circle with a center at the centroid of the Shaft Body. (See Fig. 4).

Figure 2: Indexed shaft cylinder

##  3 Robot Dynamics

###  3.1 Coordinate Frames

A generalized model of Big Foot is shown in Fig 1 and Fig 3. The components of
Big Foot include two legs with integrated feet, and a central shaft. The legs
are designed such that the center of mass is close to the ground, and the feet
give Big Foot a vertical static equilibrium orientation.

Six coordinate systems are required to fully model Big Foot’s dynamics:

  1. 1.

$\\{\boldsymbol{\hat{i}},\boldsymbol{\hat{j}},\boldsymbol{\hat{k}}\\}=\boldsymbol{I}$.

  2. 2.

$\\{\boldsymbol{\hat{i}}_{s},\boldsymbol{\hat{j}_{s}},\boldsymbol{\hat{k}}_{s}\\}=[\boldsymbol{R}_{y}(\beta)]^{T}\\{\boldsymbol{\hat{i}},\boldsymbol{\hat{j}},\boldsymbol{\hat{k}}\\}$.

  3. 3.

$\\{\boldsymbol{\hat{i}}_{1},\boldsymbol{\hat{j}}_{1},\boldsymbol{\hat{k}}_{1}\\}=[\boldsymbol{R}_{z}(\psi)]\\{\boldsymbol{\hat{i}}_{s},\boldsymbol{\hat{j}}_{s},\boldsymbol{\hat{k}}_{s}\\}$.

  4. 4.

$\\{\boldsymbol{\hat{i}}_{2},\boldsymbol{\hat{j}}_{2},\boldsymbol{\hat{k}}_{2}\\}=[\boldsymbol{R}_{x}(\phi)]\\{\boldsymbol{\hat{i}}_{1},\boldsymbol{\hat{j}}_{1},\boldsymbol{\hat{k}}_{1}\\}$.

  5. 5.

$\\{\boldsymbol{\hat{i}}_{3},\boldsymbol{\hat{j}}_{3},\boldsymbol{\hat{k}}_{3}\\}=[\boldsymbol{R}_{y}(\theta_{1})]\\{\boldsymbol{\hat{i}}_{2},\boldsymbol{\hat{j}}_{2},\boldsymbol{\hat{k}}_{2}\\}$.

  6. 6.

$\\{\boldsymbol{\hat{i}}_{4},\boldsymbol{\hat{j}}_{4},\boldsymbol{\hat{k}}_{4}\\}=[\boldsymbol{R}_{y}(\theta_{2})]\\{\boldsymbol{\hat{i}}_{2},\boldsymbol{\hat{j}}_{2},\boldsymbol{\hat{k}}_{2}\\}$.

The following coordinate transformations are required to fully define the
coordinate frames. The simple rotation transformations
$\\{[\boldsymbol{R}_{x}],[\boldsymbol{R}_{y}],[\boldsymbol{R}_{z}]\\}$ are
from Ginsberg’s foundational textbook (Ginsberg (2008)). First, a rotation of
$\beta$ about $\boldsymbol{\hat{j}}_{s}$ transforms the Space Fixed coordinate
system to the Platform Fixed coordinate system. In Fig 1 and Fig 3, the Space
Fixed coordinate system is not shown. The Platform Fixed coordinate system is
aligned with the surface Big Foot is walking on. Next, a rotation of $\psi$
about $\boldsymbol{\hat{k}}$, followed by a rotation of $\phi$ about
$\boldsymbol{\hat{j}}_{1}$, transforms the Platform Fixed coordinate system to
the Shaft Fixed coordinate system. The Platform Fixed coordinate system is
chosen as the global for the final equations of motion. Subsequently, a
rotation of $\theta_{1}$ and $\theta_{2}$ about $\boldsymbol{\hat{j}}_{2}$
transform the Shaft Fixed coordinate system to the Leg-A and Leg-B coordinate
systems. Figure 1 shows these transformations.

###  3.2 Hybrid System

As a biped robot, Big Foot is a hybrid system (Hurmuzlu et al. (2004)). In
general for bipedal walkers, there are two states $\\{a,b\\}$. These states
correspond to which foot a bipedal walker is standing on. Big Foot actually
has four states corresponding to the contact point between Big Foot and the
platform: $\\{a,b,c,d\\}$ Leg-A foot inner edge Leg-B foot inner edge Leg-A
foot curved surface Leg-B foot curved surface

These states are a result of the unique foot design. In numerical simulations,
the switching between states is achieved by toggling the respective state to 1
and the other states to 0.

###  3.3 Position Vectors

To model the kinematic constraints of motion and the centers of mass, the
following position vectors have been defined. These positions are shown in Fig
3. Point $\boldsymbol{r}_{C_{i}/O}$ is the position vector of the
foot/platform contact point. Point $\boldsymbol{r}_{A_{i}/O}$ is the position
of the center of the joint connecting the leg with the shaft. Finally,
$\boldsymbol{r}_{D/O}$ is the position of the center of the shaft.

$\boldsymbol{r}_{D/O}=\boldsymbol{r}_{A_{i}/O}+(a-b)L\boldsymbol{\hat{j}}_{2}.$
(1)

where,

$\boldsymbol{r}_{A_{i}/O}=\begin{bmatrix}X\\\ Y\\\ Z\end{bmatrix}.$ (2)

The coordinates of the center of mass for the rigid bodies are as follows. The
position vector of the center of mass (COM) of Leg-A is
$\boldsymbol{r}_{G_{A}/O}$ and Leg-B is $\boldsymbol{r}_{G_{B}/O}$. The COM of
the shaft body is $\boldsymbol{r}_{G_{D}/O}$. These vectors are defined as:

$\displaystyle\begin{split}\boldsymbol{r}_{G_{A}/O}&=\boldsymbol{r}_{D/O}-L_{G}\boldsymbol{\hat{j}}_{2}-H_{G}\boldsymbol{\hat{k}}_{3}.\\\
\boldsymbol{r}_{G_{B}/O}&=\boldsymbol{r}_{D/O}+L_{G}\boldsymbol{\hat{j}}_{2}-H_{G}\boldsymbol{\hat{k}}_{4}.\\\
\boldsymbol{r}_{G_{D}/O}&=\boldsymbol{r}_{D/O}.\end{split}$ (3)

Figure 3: The multi-body diagram of Big Foot from a frontal view

###  3.4 Angular Velocities

Next, the angular velocities are defined. The angular velocities of Leg-A,
Leg-B, and the Shaft Body are $\boldsymbol{\omega}_{1}$,
$\boldsymbol{\omega}_{2}$, and $\boldsymbol{\omega}_{3}$, respectively.

$\displaystyle\begin{split}\boldsymbol{\omega}_{1}&=\dot{\psi}\boldsymbol{\hat{k}}+\dot{\phi}\boldsymbol{\hat{i}}_{1}+\dot{\theta}_{1}\boldsymbol{\hat{j}}_{2}.\\\
\boldsymbol{\omega}_{2}&=\dot{\psi}\boldsymbol{\hat{k}}+\dot{\phi}\boldsymbol{\hat{i}}+\dot{\theta}_{2}\boldsymbol{\hat{j}}_{2}.\\\
\boldsymbol{\omega}_{3}&=\dot{\psi}\boldsymbol{\hat{k}}+\dot{\phi}\boldsymbol{\hat{i}}_{1}.\end{split}$
(4)

There is only two angular velocities for the Shaft Body because there are no
moments acting about the $\boldsymbol{\hat{j}}_{2}$ direction, which makes one
degree of freedom of the Shaft Body static and able to be ignored.

###  3.5 Velocity Constraint

Next, a velocity constraint must be defined for the position of Big Foot with
respect to the walking surface. The feet are designed to be circular, as shown
in Fig 4.

Figure 4: Side view of foot curvature

Two velocity constraints exist for each state. The first velocity constraint
is the requirement that the foot is in contact with the ground. This is
modeled as the velocity of the foot contact point with the platform,
$\dot{C}_{i}$, is zero.

Next, a velocity constraint exists that represents the roll of the circular
foot on the platform as Big Foot moves. There are four equations for this
constraint corresponding to each state.

$\boldsymbol{\dot{r}}_{A_{i}/O}=\boldsymbol{\omega}_{1}\times\boldsymbol{r}_{A_{A}/C_{A}}$
$\boldsymbol{\dot{r}}_{A_{i}/O}=\boldsymbol{\omega}_{2}\times\boldsymbol{r}_{A_{B}/C_{B}}$
$\boldsymbol{\dot{r}}_{A_{i}/O}=\boldsymbol{\omega}_{1}\times(H_{B}\boldsymbol{\hat{k}})-L\boldsymbol{\dot{\hat{j}}_{2}}$
$\boldsymbol{\dot{r}}_{A_{i}/O}=\boldsymbol{\omega}_{2}\times(H_{B}\boldsymbol{\hat{k}})+L\boldsymbol{\dot{\hat{j}}_{2}}$
where,

$\boldsymbol{r}_{A_{i}/C_{i}}=H\boldsymbol{\hat{k}}_{2}.$ (5)

and $H_{B}$ is the radius of the spherical surface of the feet.

The $\boldsymbol{\hat{k}}$-direction of the rolling constraint is holonomic
and when integrate for each state is: $Z=H\cos{\phi}$ $Z=H\cos{\phi}$
$Z=H_{B}-L\sin{\phi}$ $Z=H_{B}+L\sin{\phi}$

###  3.6 Energy Definitions

Now that the position vectors have been fully defined, the Lagrangian can be
derived. We start with the different types of energy. First, we have
translational kinetic energy:

$\begin{split}T_{t}=&\frac{1}{2}(m_{A}\
\boldsymbol{\dot{r}}_{G_{A}/O}\cdot\boldsymbol{\dot{r}}_{G_{A}/O}+\\\ &m_{B}\
\boldsymbol{\dot{r}}_{G_{B}/O}\cdot\boldsymbol{\dot{r}}_{G_{B}/O}+m_{C}\
\boldsymbol{\dot{r}}_{G_{C}/O}\cdot\boldsymbol{\dot{r}}_{G_{C}/O})\end{split}$
(6)

where, $m_{A}$, $m_{B}$, and $m_{C}$ are the masses of the two legs and the
shaft body, respectively. Next, the rotational kinetic energy:

$\begin{split}T_{r}=&\frac{1}{2}(\boldsymbol{\omega}_{1}^{T}[\boldsymbol{R}_{3}]^{T}\boldsymbol{I}_{1}[\boldsymbol{R}_{3}]\boldsymbol{\omega}_{1}+\\\
&\boldsymbol{\omega}_{2}^{T}[\boldsymbol{R}_{4}]^{T}\boldsymbol{I}_{2}[\boldsymbol{R}_{4}]\boldsymbol{\omega}_{2}+\boldsymbol{\omega}_{3}^{T}[\boldsymbol{R}_{2}]^{T}\boldsymbol{I}_{3}[\boldsymbol{R}_{2}]\boldsymbol{\omega}_{3})\end{split}$
(7)

where, $\boldsymbol{I}_{1}$ is the inertia matrix of Leg-A with respect to
$\\{\boldsymbol{\hat{x}}_{3},\boldsymbol{\hat{y}}_{3},\boldsymbol{\hat{z}}_{3}\\}$,
$\boldsymbol{I}_{2}$ is the inertia matrix of Leg-B with respect to
$\\{\boldsymbol{\hat{x}}_{4},\boldsymbol{\hat{y}}_{4},\boldsymbol{\hat{z}}_{4}\\}$,
$\boldsymbol{I}_{3}$ is the inertia matrix of the Shaft Body with respect to
$\\{\boldsymbol{\hat{x}}_{2},\boldsymbol{\hat{y}}_{2},\boldsymbol{\hat{z}}_{2}\\}$,
and the rotation matrices are defined below:

$\displaystyle\begin{split}[\boldsymbol{R}_{2}]&=\begin{bmatrix}\boldsymbol{\hat{i}}_{2}^{T}\\\
\boldsymbol{\hat{j}}_{2}^{T}\\\ \boldsymbol{\hat{k}}_{2}^{T}\end{bmatrix}\\\
[\boldsymbol{R}_{3}]&=\begin{bmatrix}\boldsymbol{\hat{i}}_{3}^{T}\\\
\boldsymbol{\hat{j}}_{3}^{T}\\\ \boldsymbol{\hat{k}}_{3}^{T}\end{bmatrix}\\\
[\boldsymbol{R}_{4}]&=\begin{bmatrix}\boldsymbol{\hat{i}}_{4}^{T}\\\
\boldsymbol{\hat{j}}_{4}^{T}\\\
\boldsymbol{\hat{k}}_{4}^{T}\end{bmatrix}.\end{split}$ (8)

Finally, the potential energy of the system is:

$V=g(m_{A}\boldsymbol{r}_{G_{A}/O}+m_{B}\boldsymbol{r}_{G_{B}/O}+m_{C}\boldsymbol{r}_{G_{C}/O})\cdot\boldsymbol{\hat{K}}.$
(9)

The Lagrangian can be written as:

$\mathcal{L}=T_{t}+T_{r}-V,$ (10)

###  3.7 Constraint Forces

The constrained generalized coordinates for the system are:

$\boldsymbol{q}=\\{\theta_{1},\theta_{2},\phi,\psi,x,y\\}.$ (11)

As a result of the rolling velocity constraint, there are two constraint
forces $\\{r_{x},r_{y}\\}$. These forces interact with the system through
generalized forces and moments ($Q_{ic}$):

$\displaystyle\begin{split}Q_{1c}=&-(a\ H+c\
H_{B}\cos{\phi})(r_{x}\cos{\psi}+r_{y}\sin{\psi})\\\ Q_{2c}=&-(b\ H+d\
H_{B}\cos{\phi})(r_{x}\cos{\psi}+r_{y}\sin{\psi})\\\
Q_{3c}=&(r_{y}\cos{\psi}-r_{x}\sin{\psi}(H(a+b)\cos{\phi}+\\\
&H_{B}(c+d)+L(d-c)\sin{\phi})\\\
Q_{4c}=&(r_{x}(-\cos{\psi})-r_{y}\sin{\psi})(H(a+b)\sin{\phi}+\\\
&L(c-d)\cos{\phi})\\\ Q_{5c}=&r_{x}\\\ Q_{6c}=&r_{y}\end{split}$ (12)

There are two types of external loads. The first type of external load is
friction. There is rolling friction between the feet and the platform (Szirtes
and Rózsa (2007)):

$\displaystyle\begin{split}Q_{1r}&=-c_{r1}sgn(\dot{\theta}_{1})\\\
Q_{2r}&=-c_{r1}sgn(\dot{\theta}_{2})\\\ Q_{3r}&=-c_{r2}sgn(\dot{\phi})\\\
Q_{4r}&=-c_{r2}sgn(\dot{\psi})\end{split}$ (13)

where, $c_{ri}$ is the rolling coefficient for the respective dynamics.

The model also includes the normal force between the foot and the platform,
but this normal force varies insignificantly and can be assumed to be
constant.

The damping friction term is:

$\displaystyle\begin{split}Q_{1f}&=-c_{f1}\dot{\theta}_{1}\\\
Q_{2f}&=-c_{f1}\dot{\theta}_{2}\end{split}$ (14)

where, $c_{fi}$ is the damping coefficient for the respective dynamics. The
damping friction of $\theta_{i}$ is the result of the constraint forces
between the legs and the Shaft Body.

###  3.8 Magnetic Moment

The active control of Big Foot is achieved through an arbitrary uniform
magnetic field (Al Khatib et al. (2020)). The magnetic field is generated with
3 sets of orthogonal Helmholtz coils. The magnetic field produced by in the
inner set of coils is shown in Fig. 5. The yellow rectangle represents the
foam platform the experiments are run on. After calibration, the magnetic
field can be represented by a vector $p_{m}\boldsymbol{\hat{j}}_{m}$, where
$p_{m}$ is the power level of the field and $\boldsymbol{\hat{j}}_{m}$ is the
$y$-direction of the coordinate frame as defined below:

$\\{\boldsymbol{\hat{i}}_{m},\boldsymbol{\hat{j}}_{m},\boldsymbol{\hat{k}}_{m}\\}=[\boldsymbol{R}_{x}(\phi_{m})][\boldsymbol{R}_{z}(\psi_{m})]\boldsymbol{\\{\hat{i},\hat{j},\hat{k}\\}}$
(15)

Figure 5: Cross-section view of magnetic field generated by the inner
Helmholtz coils.

Once the magnetic field has been created, the magnet in Big Foot interacts
with the field with the following magnetic moment equation:

$\boldsymbol{\tau}_{m}=(k_{m}\boldsymbol{\hat{j}}_{2})\times(p_{m}\boldsymbol{\hat{j}}_{m})$
(16)

where, $k$ is a magnet strength coefficient found through tuning the model to
experimental results.

Next, to find the applied load of the magnetic moment on the shaft assembly’s
generalized coordinates, the moment is projected on each coordinate’s rotation
axis:

$\displaystyle\begin{split}Q_{3m}=&\boldsymbol{\tau}_{m}.\boldsymbol{\hat{i}}_{1}\\\
=&k_{m}p_{m}(\sin{\phi_{m}}\cos{\phi}-\cos{\phi_{m}}\sin{\phi}\cos{(\psi_{m}-\psi)})\\\
Q_{4m}=&\boldsymbol{\tau}_{m}.\boldsymbol{\hat{k}}\\\
=&k_{m}p_{m}\cos{\phi_{m}}\cos{\phi}\sin{(\psi_{m}-\psi)}\end{split}$ (17)

Finally, using the Lagrangian method, the equations of motion for Big Foot
are:

$\displaystyle\begin{split}\frac{d}{dt}\Bigl{(}\frac{\partial\mathcal{L}}{\partial\dot{\theta}_{1}}\Bigr{)}-\frac{\partial\mathcal{L}}{\partial\theta_{1}}&=Q_{1c}+(a+c)\
Q_{1r}+Q_{1f}\\\
\frac{d}{dt}\Bigl{(}\frac{\partial\mathcal{L}}{\partial\dot{\theta}_{2}}\Bigr{)}-\frac{\partial\mathcal{L}}{\partial\theta_{2}}&=Q_{2c}+(b+d)\
Q_{2r}+Q_{2f}\\\
\frac{d}{dt}\Bigl{(}\frac{\partial\mathcal{L}}{\partial\dot{\phi}}\Bigr{)}-\frac{\partial\mathcal{L}}{\partial\phi}&=Q_{3c}+Q_{3r}+Q_{3m}\\\
\frac{d}{dt}\Bigl{(}\frac{\partial\mathcal{L}}{\partial\dot{\psi}}\Bigr{)}-\frac{\partial\mathcal{L}}{\partial\psi}&=Q_{4c}+Q_{4r}+Q_{4m}\\\
\frac{d}{dt}\Bigl{(}\frac{\partial\mathcal{L}}{\partial\dot{x}}\Bigr{)}-\frac{\partial\mathcal{L}}{\partial
x}&=Q_{5c}\\\
\frac{d}{dt}\Bigl{(}\frac{\partial\mathcal{L}}{\partial\dot{y}}\Bigr{)}-\frac{\partial\mathcal{L}}{\partial
y}&=Q_{6c}\end{split}$ (18)

.

###  3.9 Impact Map

Every time Big Foot switches stance legs, there is an impact. This impact is
observed through a loss of kinetic energy and audible sound produced.

Hurmuzlu (2020) notes that the Lagrangian formalism for impact problems can be
written as:

$\begin{split}\Biggl{[}\frac{\partial
T}{\partial\dot{q}_{i}}\Biggl{]}^{+}-\Biggl{[}\frac{\partial
T}{\partial\dot{q}_{i}}\Biggl{]}^{-}-\hat{Q}_{i}-\sum_{K}^{j=1}\lambda_{j}\frac{\partial\psi_{j}}{\partial\dot{q}_{i}}=0\\\
i=1,2,...,N+2\end{split}$ (19)

where, $T$ is the total kinetic energy, the superscripts $-$ and $+$
correspond to the states immediately before and after impact, respectively,
$\partial\dot{q}_{i}$ is the $i$th generalized velocity, $\hat{Q}_{i}$ is the
generalized impulse in the direction of the $i$th generalized coordinate, and
$\psi_{j}$ and $\lambda_{j}$ are the $j$th constraint and the corresponding
Lagrange multiplier, respectively.

In using this impact model, once $\bigl{[}\frac{\partial
T}{\partial\dot{q}_{i}}\bigr{]}^{+}$ and $\bigr{[}\frac{\partial
T}{\partial\dot{q}_{i}}\bigr{]}^{-}$ are evaluated, the velocity constraints
are substituted back in. This eliminates the constrained coordinates from the
impact model and simplifies the map. Also, to model the movement of the
foot/platform contact point of the pre-impact stance leg, the generalized
coordinates are expanded as follows:

$\boldsymbol{q}_{impacts}=\\{\theta_{1},\theta_{2},\phi,\psi,c_{x},c_{y},c_{z}\\}.$
(20)

The position of these contact points is incorporated by defining the velocity
constraint at impact as:

$\boldsymbol{\dot{r}}_{A_{i}/O}=\boldsymbol{\dot{c}}_{i}+a(\boldsymbol{\omega}_{1}\times\boldsymbol{r}_{A_{A}/C_{A}})+b(\boldsymbol{\omega}_{2}\times\boldsymbol{r}_{A_{B}/C_{B}})$
(21)

where, $\boldsymbol{\dot{c}}_{i}$ is the velocity of point $c_{i}$ and state
$a$ is used when the stance leg switches from Leg-a to Leg-b and visa versa.

The generalized impulse, $\hat{Q}_{i}$, is found as:

$\displaystyle\begin{split}Q_{1c}&=(a-b)H(\tau_{x}\cos{\psi}+\tau_{y}\sin{\psi})\\\
Q_{2c}&=(b-a)H(\tau_{x}\cos{\psi}+\tau_{y}\sin{\psi})\\\ Q_{3c}&=2(a-b)L\
\tau_{z}\\\ Q_{4c}&=-2(a-b)L(\tau_{x}\cos{\psi}+\tau_{y}\sin{\psi})\\\
Q_{5c}&=\tau_{x}\\\ Q_{6c}&=\tau_{y}\\\ Q_{7c}&=\tau_{z}\end{split}$ (22)

where, $\tau_{i}$ is the contact impulses at the striking foot and $\phi$ is
set equal to 0 because this is the stance switching angle.

Finally, we have a set of equations to solve. There is the Lagrangian
formalization from Eq. 19 with the generalized impulses in Eq. 22 and the
substituted impact velocity constraints in Eq. 21. Next, there are post- and
pre-impact velocity constraints. Before impact, the contact point on the
stance leg is static, meaning:

$\boldsymbol{\dot{c}}^{-}=0$ (23)

After impact, the post-impact stance contact point is static. This is
represented kinematically as:

$\displaystyle\begin{split}\dot{c}_{x}^{+}&=-\Bigl{(}(a-b)\cos{\psi}\bigl{(}H\dot{\theta}_{1}^{+}-H\dot{\theta}_{2}^{+}-2L\dot{\psi}^{+}\bigr{)}\Bigr{)}\\\
\dot{c}_{y}^{+}&=(a-b)\sin{\psi}\bigl{(}-H\dot{\theta}_{1}^{+}+H\dot{\theta}_{2}^{+}+2L\dot{\psi}^{+}\bigr{)}\\\
\dot{c}_{z}^{+}&=2L(b-a)\dot{\phi}^{+}\end{split}$ (24)

This post impact constraint comes from assuming the coefficient of restitution
is 0. Next, these equations are fed into a numerical solver and the post-
impact velocities can be found through the solution. This solution is called
the impact map.

Once the equations of motion and the impact map has been derived, the system
can be simulated.

##  4 Actuation Schemes

We tailor the actuation such that we can study the following questions
regarding the application of the magnetic torque:

  1. 1.

The timing of the external torque, whether it should be applied at the end of
the heel strike or whether it should be in the form of an external beat and
let the biped adjust to it.

  2. 2.

The nature of the pulse, whether it should be in the form of high-magnitude
short duration (impact like) pulse or longer duration but lower magnitude
pulse.

During both approaches a magnetic field in the form of square wave is
considered. The applied field has five control parameters:

  * •

$\psi_{m}$, is the yaw angle of the magnetic field vector.

  * •

$\phi_{m}$, is the roll angle of the magnetic field vector.

  * •

$P_{L}$, is the time duration of the applied magnetic field.

  * •

$P_{m}$, is the height of the applied magnetic field.

  * •

$P_{Area}$ is the magnitude of the applied magnetic impulse

where, $P_{m}$ is unitless. $P_{m}$ is a multiplier of a magnetic field power
defined in the equations of motion.

The rotational impulse acting on the biped by the magnetic field is defined
as:

$I_{m}=\int_{t_{i}}^{t_{i}+P_{L}}||\boldsymbol{\tau}_{m}||\,dt$ (25)

where, $t_{i}$ and $t_{i}+P_{L}$ are the time of onset and termination of the
magnetic pulse respectively. The consequence of considering a square pulse is:

$P_{Area}=P_{L}\times P_{m}$ (26)

with $P_{Area}$, regimes can be defined that correspond to the interaction of
the input magnetic field with the dynamical response.

I. Impact regime :

This can be described as a Very short duration and a high magnitude pulse.
During this regime, the generalized coordinates and velocities remain
relatively unchanged during application of the magnetic torque. Thus, the
applied magnetic torque has a similar effect to that of an external impact.

II. Impulsive regime:

This can be described as a moderate magnitude and duration pulse. During this
regime, the generalized coordinates and velocities change significantly during
the application of the magnetic torque. Thus, the applied magnetic torque
cannot be characterized as an external impact.

Responses illustrating these regimes are shown in Figs. 6(a) and 6(b).

In addition, as mentioned before, two actuation timing approaches will be
considered.

I. Heel strike based actuation:

The first scheme is based on triggering the pulse immediately after the heel
strike:

$\phi[t]=\begin{cases}\phi_{in}&\phi>0\\\
-\phi_{in}&\phi<0\end{cases}\;\;\;\text{,}$ (27)

$\psi[t]=\begin{cases}\psi_{in}&\phi>0\\\
-\psi_{in}&\phi<0\end{cases}\;\;\;\text{, and}$ (28)

$P_{m}=\begin{cases}P_{in}&t_{s}<t<t_{s}+P_{L}\\\ 0\end{cases}$ (29)

where, $t_{s}$ is the time of heel strike. An example run of this foot strike
actuation is shown in Fig. 6(a) and Fig. 6(b).

II. Constant period actuation

The second actuation scheme is completely open-loop and is simply a pattern of
periodically applied pulses:

$\phi_{m}=\begin{cases}\phi_{in}&0<\text{mod}{(t,t_{4})}<t_{1}\\\
-\phi_{in}&t_{2}<\text{mod}{(t,t_{4})}<t_{3}\end{cases}\;\;\;\text{,}$ (30)

$\psi_{m}=\begin{cases}\psi_{in}&0<\text{mod}{(t,t_{4})}<t_{1}\\\
-\psi_{in}&t_{2}<\text{mod}{(t,t_{4})}<t_{3}\end{cases}\;\;\;\text{, and}$
(31)

$P_{m}=\begin{cases}P_{in}&(0<\text{mod}{(t,t_{4})}<t_{1})||(t_{2}<\text{mod}{(t,t_{4})}<t_{3})\\\
0\end{cases}$ (32)

where, $t_{1}=P_{L}$, $t_{2}=P_{L}+t_{off}$, $t_{3}=2P_{L}+t_{off}$,
$t_{4}=2P_{L}+2t_{off}$, and $t_{off}$ is the time period that the pulse is
turned off.

An example run of this constant pulse wave actuation is shown in Fig. 6(c).

(a) Heel strike actuation scheme in impact regime

(b) Heel strike actuation scheme in impulsive regime

(c) Constant pulse wave actuation scheme in impulsive regime

Figure 6: The two actuation schemes in different regimes

In Fig. 6, the plots show steady state responses. Notice the timing between
the heel strike and the switching of the input pulse. For the heel strike
actuation, as enforced by the magnetic torque input, the heel strike and the
input pulse are synchronous. But, for the constant pulse wave, the pulse does
not necessarily initiate at the heel strike. This shows how the gait pattern
can be initiated without the syncing of the pulse wave and the heel strike.
The selection of actuation parameters depend on the desired gait
characteristics.

##  5 Stability Analysis

Let
$\phi_{p}(t)=(\theta_{1},\theta_{2},\phi,\psi,\dot{\theta}_{1},\dot{\theta}_{2},\dot{\phi},\dot{\psi})\in\mathbb{R}^{8}$
be a periodic solution of Eq. 18 (Kashki et al. (2017)). The one-sided hyper
plane Poincaré section is defined as

$\mathcal{S}\equiv(\theta_{1},\theta_{2},\psi,\dot{\theta}_{1},\dot{\theta}_{2},\dot{\phi},\dot{\psi})\in\mathbb{R}^{7}:\phi=0,\dot{\phi}<0.$
(33)

If $\boldsymbol{\xi}[k]\in\mathcal{S}$ denotes the $k$th intersection of
$\mathcal{S}$ by the flow of $\phi_{p}$, the discrete-time Poincaré Map
$\mathcal{P}:\mathcal{S}\rightarrow\mathcal{S}$ can be expressed as

$\boldsymbol{\xi}[k+1]=\mathcal{P}(\boldsymbol{\xi}[k])$ (34)

Subsequently, if $\boldsymbol{\xi}^{*}$ stands for a fixed point of the
Poincaré Map, then the local exponential stability of $\boldsymbol{\xi}^{*}$
on $\mathcal{S}$ is equivalent to local exponential stability of the
underlying limit cycle (Westervelt et al. (2007)). We used Floquet Theory for
stability analysis of a specific limit cycle (Hurmuzlu and Moskowitz (1987a);
Hurmuzlu and Moskowitz (1987b)). Therefore, the local linearization of the
Poincaré Map about $\boldsymbol{\xi}^{*}$ gives:

$\displaystyle\begin{split}\mathcal{P}(\boldsymbol{\xi})\simeq&\mathcal{J}(\boldsymbol{\xi}^{*})(\boldsymbol{\xi}-\boldsymbol{\xi}^{*})\\\
\mathcal{J}(\boldsymbol{\xi})=&\frac{\partial\mathcal{P}(\boldsymbol{\xi})}{\partial\boldsymbol{\xi}}\end{split}$
(35)

where $\mathcal{J}(\boldsymbol{\xi})$ is the $7\times 7$ linearized Jacobian
matrix of $\mathcal{P}(\boldsymbol{\xi})$. Next, the Floquet multipliers are
defined as:

$\rho_{i}=|Re(\lambda_{i})|:\lambda_{i}=\text{eig}(\mathcal{J}(\boldsymbol{\xi}))$
(36)

where $\rho_{i}$ and $\lambda_{i}$ are the $i$th Floquet multiplier and
eigenvalue of $\mathcal{J}(\boldsymbol{\xi})$. Accordingly, stability of the
limit cycle can be defined as:

$\phi_{p}(t)=\begin{cases}\text{stable}:&\forall\rho_{i}<1\\\
\text{unstable}:&\exists\rho_{i}\geq 1\end{cases}$ (37)

The algorithm used to find the Floquet multipliers is shown below:

Define list $\boldsymbol{A}=\\{\\}$

for  _$n\in\mathcal{S}$_ do

$\mathcal{S}(0)_{n}=(1+\delta)\mathcal{S}(0)_{n}$

Integrate for one cycle

Join $\boldsymbol{A}$ and
$\frac{\mathcal{S}^{*}-\mathcal{S}(t_{f})}{\delta\mathcal{S}(0)_{n}}$ to form
a new $\boldsymbol{A}$, where $t_{f}$ is the final time of the cycle.

end for

$\rho_{i}=||\text{eig}(\boldsymbol{A})_{i}||$

Algorithm 1 Floquet Multiplier Finder

Next, we explore the effect of the input parameter values on the resulting
gait patterns.

##  6 Simulations

The simulations were conducted using Wolfram Research Mathematica software. In
the next subsections we present the simulation results. We study the dynamic
behavior for the two actuation schemes considering the following resulting
outcomes:

  1. 1.

Existence and type of resulting limit cycles.

  2. 2.

Stability of the period-1 limit cycles.

  3. 3.

Step length and progression speed of the biped.

We vary the following parameters when we conduct the simulations:

  1. 1.

Pulse duration $P_{L}$

  2. 2.

Pulse area $P_{Area}$

  3. 3.

Walking surface inclination $\beta$

We have two objectives here: a) first, is to identify the parameter ranges
that result in the most stable period-1 cycles with the highest possible
progression speed over the steepest walking surface, b) second, is to compare
the dynamic response for impact and impulsive regimes.

###  6.1 Heel Strike Scheme Simulation Results

In Fig. 7, the results of the simulations for the heel strike actuation scheme
are shown. The results are composed of 3 sets of different $P_{L}$
values:$\\{5\text{ ms},22.5\text{ ms},40\text{ ms}\\}$. In each set five pulse
areas were used:$\\{2\text{ ms},4\text{ ms},6\text{ ms},8\text{ ms},10\text{
ms}\\}$. For simplicity, the direction of the applied magnetic pulse is kept
constant at $\\{\phi_{m}=60^{\circ},\psi_{m}=20^{\circ}\\}$. The simulations
were initiated by running the model at a $0^{\circ}$ slope for each $P_{Area}$
until either a limit cycle is identified, a pre-determined simulation time has
expired, or locomotion is lost. Once the steady state locomotion is identified
for a flat slope, the nature of the gait is added to the bifurcation map, the
average maximum Floquet multiplier is calculated, and the slope is incremented
by $0.5^{\circ}$ for the next run. The increasing of slope continued until
either the robot begins walking backward or locomotion is lost.

In algorithm 1, the calculation of the Floquet multipliers is explained. The
perturbation, $\delta$, is varied between 0.01 and 0.1. Then, for each
$\delta$, the maximum Floquet multiplier is selected. Finally, the average of
the maximum Floquet multipliers for each $\delta$ is calculated to represent
the average local stability of the fixed point. This value is printed next to
the corresponding sample point on Fig. 7 and Fig. 9. From the map, the maximum
slope the robot can walk and the corresponding gait patterns and stability has
been identified.

Figure 7: Forward gait patterns in heel strike actuation

As stated previously, we conduct the simulations in order to identify the
optimal operating region for Big Foot. We have chosen three goals for
locomotion. The first goal is to walk the steepest slope. Based on our
results, longer pulse lengths with higher pulse areas is best (larger
$P_{m}$). In our runs, a maximum slope of $4.5^{\circ}$ at a pulse length of
$40\text{ ms}$ and a pulse area of $10\text{ ms}$ is achieved. The second goal
is to walk the steepest slope with a period-1 gait. This occurred at
$4^{\circ}$ at the same parameter values. The period-1 gait is useful for
controllability as the displacement per step remains unchanged during
locomotion. The third goal is to achieve maximum stability. Based on the
average maximum Floquet multipliers, smaller pulse areas result in higher
stability.

Several overall trends in the results were observed as well. Quasi-Periodic
gaits were observed as, while increasing the slope, the gait transitioned from
one periodic gait to another. In Fig. 8, the Poincaré return map for $\psi$ on
a Quasi-Periodic orbit is shown as an example. For $P_{L}=5\text{ ms}$ and
$P_{L}=22.5\text{ ms}$, the bifurcation plot have similar trends as $P_{Area}$
is increased. Initial, for $P_{L}=5\text{ ms}$ and $P_{L}=22.5\text{ ms}$, as
$P_{Area}$ is increased, the maximum slope increases. The maximum for these
ranges of $P_{Area}$ is observed as the point to which Big Foot would start
walking backwards. For $P_{L}=5\text{ ms}$ at $P_{Area}=6\text{ ms}$ and for
$P_{L}=22.5\text{ ms}$ at $P_{Area}=8\text{ ms}$, this point switches to being
the point of instability. Meaning, for $P_{Area}$ greater than this point, the
maximum slope is limited by stability. At these $P_{Area}$, Big Foot falls and
loses locomotion at steeper slopes.

Figure 8: Poincaré Return Map for the yaw angle in a Quasi-Periodic gait.

###  6.2 Constant Pulse Wave Scheme Simulation Results

Next, simulations for the constant pulse wave actuation scheme were conducted.
A bifurcation map of the same structure of Fig. 7 is setup for the constant
pulse wave scheme. The results of this map are shown in Fig. 9. For this map,
the same parameters of Fig. 7 with a $t_{off}=60\text{ ms}$ is chosen.

Figure 9: Forward gait patterns in constant pulse wave actuation

The difference in results of the heel strike scheme and the constant pulse
wave scheme are as follows. The maximum slope achieved by the constant pulse
wave scheme is significantly higher with $6^{\circ}$ at $P_{L}=40\text{ ms}$
and $P_{Area}=10\text{ ms}$. Another difference is the generation of different
periodic gaits. The heel strike scheme predictably performed period-1 gaits at
lower slopes. Whereas, the gait patterns of the constant pulse wave do not
appear to be predictable from this study. For the constant pulse wave scheme a
gait pattern called ”Non-Periodic” is identified. ”Non-Periodic” refers to a
gait that appeared to have a random gait pattern and could not be classified
as periodic or quasi-periodic. Another observation is that the performance did
not vary greatly with different pulse lengths for the constant pulse wave,
where it does for the heel strike. For the constant pulse wave, the robot is
only operational in pulse areas $6\text{ ms}$ through $10\text{ ms}$ for all
pulse lengths. For a control engineer, the heel strike appeared to be more
desirable period-1 gait, but the constant pulse wave scheme achieves
significantly greater slopes. Another difference is stability. For the
constant pulse wave with $P_{L}=22.5\text{ ms}$, lower pulse areas were found
to be less stable. This is believed to be a result of the same dynamics that
cause the lack of predictability in gait pattern as the generation of
different gait patterns developed at instability.

In Fig. 10, the maximum slopes achieved in the simulations are shown. These
plots show the behavior of the performance as you change actuation schemes and
input parameters. For the heel strike scheme, the maximum achieved slope
appears to increase as you increase the pulse length. However, for the
constant pulse wave actuation, the maximum slope occured at $P_{Area}=10\text{
ms}$ for all pulse durations. The maximum achieved slope for the constant
pulse wave scheme increased with larger pulse areas.

Figure 10: Maximum Slopes Achieved

Another performance metric of Big Foot is the locomotion travel, or more
specifically the velocity and stride length of the gaits. In Fig. 11 and Fig.
12, these metrics are presented. To find the velocity and stride length, each
fixed-point is run for 10 seconds. Then, the velocity is calculated by
dividing the distance covered by 10 seconds. The stride length is calculated
by dividing the distance covered by the number of steps taken. These
calculations provide averages, which are required as many of the fixed-points
were not period-1.

The results of the study of the travel show the velocities and stride lengths
match in their relationship to slope, pulse area, and pulse duration. Also,
the velocities and stride lengths appear to be independent of the gait
patterns generated. Finally, the highest travel velocity (50 mm/s) and stride
length (8mm) are achieved by the heel strike actuation scheme at
$P_{l}=22.5\text{ ms}$ and $P_{Area}=10\text{ ms}$.

Figure 11: Simulation velocities

Figure 12: Simulation Stride Lengths

##  7 Experimental Setup

The experimental setup includes 3 pairs of Helmholtz coils, a test platform, a
light, a camera, a Microcontroller, a motor driver, and a PC. Each pair of
Helmholtz coils requires two identical, parallel coils separated by a distance
equal to their radius. Both coils are wired such that their current runs in
the same direction. The experimental setup follows the outline set by Abbott
(2015) for nested circular coils that was used by Al Khatib et al. (2022) as
well. The setup involves three pairs of Helmholtz coils positioned
orthogonally around a platform on which Big Foot walks, as pictured in Fig. 13

Figure 13: Experimental setup of three orthogonally positioned pairs of
Helmholtz coils

The outer diameters of the $y$, $x$, and $z$ coils are approximately 46 cm, 40
cm, and 31 cm respectively. The inner diameters are approximately 40 cm, 34
cm, and 25.5 cm and the coil thicknesses are approximately 4.47 cm, 3.61 cm,
and 3.21 cm. The platform is machined out of engineering foam, and an Ultra-
Thin 10A durometer Silicone Rubber Sheet is placed on top of the platform’s
surface to prevent slipping. At the top of the setup a Razer Kiyo Pro camera
is secured with a XJ-19 Selfie Ring Light around it. Reflective tape is
applied to the Shaft Body of Big Foot. Testing of various incline angles is
done with the use of 3D printed ramps that hooked on the bottom $z$ coil.

The approximately uniform magnetic fields caused by the three pairs of coils
induce torques, as briefly noted in Eq. 16. The input control of the coils
contains three variables that are the attitude angles and power of the overall
created magnetic field. These variables are: yaw ($\psi_{m}$), pitch
($\phi_{m}$), and power($P$). The equations used to convert these inputs into
actual coil voltages are given as follows:

$\begin{split}P_{x}&=P\cos{\phi_{m}}\cos{\psi_{m}}k_{x}\\\
P_{y}&=P\cos{\phi_{m}}\sin{\psi_{m}}k_{y}\\\
P_{z}&=P\sin{\phi_{m}}k_{z}\end{split}$ (38)

where, $\\{P_{x},P_{y},P_{z}\\}$ are the separate voltages applied to each
orthogonal coil and $\\{k_{x},k_{y},k_{z}\\}$ are the correction factors that
account for unit conversions and intermediate experimental parameters.

###  7.1 Big Foot Prototype

Figure 14: The Big Foot

The prototype of Big Foot is shown in Fig. 14. The Shaft Cylinder is machined
from Delrin. The collars and legs were laser cut using the PS48 laser cutter
and engraver. The leg is 6.5x0.79x1.75 mm. The feet are 3D printed using a
Photon S SLA 3D printer. The feet are a section of an 8mm sphere, which are
glued to the legs. The shaft is a 0.794mm non-magnetic stainless steel shaft.
The magnets used are 1mm diameter Neodymium magnets.

##  8 Experimental Results

A set of walking tasks is defined to validate the performance of the Big Foot
prototype. First, Big Foot is placed along a straight path with different
slopes to compare the prototype’s dynamics to the model. Second, a variety of
two dimensional paths are prescribed with Big Foot to demonstrate maneuvering
capabilities.

###  8.1 Uphill Walking Experiments

To compare the response of the Big Foot prototype to the model, the prototype
is placed on a series of slopes. As no sensors were available to detect the
heel strikes, experiments are limited to the constant pulse wave actuation
scheme. The parameters of this test were as
follows:$\\{\psi_{m}=20^{\circ},\phi_{m}=60^{\circ},P_{m}=33\%,P_{L}=300\text{
ms},T_{off}=60\text{ ms}\\}$. A long pulse width is used because a very short
pulse lengths are not possible because the cut off frequency of the largest
Helmholtz coil is found experimentally to be 5Hz. Video footage of the
experiment is shown in Extension 1.

Figure 15: Experimental results compared to Simulations.

Figure 15 shows the stride length’s variation with respect to the walking
surface slope for both experiments and simulations. The simulation results
agree with the experiments. Experimental outcomes are similar to the
theoretical ones. The stride length is greater in the experiment, but this can
be explained in experimental uncertainties such as manufacturing tolerances
and inconsistent surface contact. However, the result in Fig. 15 are
consistent with other runs and show the same relationship in experiments and
simulations. In constant pulse wave actuation, the stride length, and
consequentially the forward velocity, monotonically decreases as the walking
surface slope increases.

The next experiment is designed to capture the effect of pulse frequency on
the stride length. In the experiment with results in Fig. 16, the prototype
walks across a flat surface. The input pulse duration is incremented by 10%
between each run.

Figure 16: Stride Length with respect to input frequency

We observe two different gait characteristics at two extremes of the input
frequencies. For high frequencies the gait is more dynamic because of the
short duration pulses. For low frequencies, however, the gait becomes more
quasi-static. This is due to the long duration pulse where the biped becomes
almost static towards the end of each step cycle. When the contra-lateral
pulse arrives, the biped transitions to the next quasi-static posture,
achieving forward locomotion. For this reason, for lower frequency values,
step length remains almost unchanged. There is a clear transition from
impulsive actuation to continuous actuation as the input frequency exceeds a
certain value (approximately 1.4 Hz). Experimentally we observe that the
longest step lengths are achieved at the highest input frequencies.
Consequently, we observe that impulsive actuation produces longer step lengths
than continuous actuation.

###  8.2 Miscellaneous Maneuvers

In this paper, so far we have focus on locomotion along a single direction. It
is possible to walk in two dimensions by adding a new input parameter through
the following substitution:

$\psi_{m}\rightarrow\psi_{m}+\psi_{d}$ (39)

where, $\psi_{d}$ is a direction angle.

Big Foot can be steered using $\psi_{d}$. Each time a pulse is administered a
magnetic field is generated in a given direction. This results in an applied
moment vector in that direction. Hence, the locomotion direction can be
altered by varying this $\psi_{d}$ angle. In order to demonstrate this
feature, four sets of maneuvers were performed. These sets can be viewed in
Extension 2-5. Figure 17 depicts the four tasks that we put into Big Foot.

(a) Square Path

(b) Circular Path

(c) Figure-8 Path

(d) Lego® Maze

Figure 17: 2D Locomotion Paths

The first maneuver is programmed to have Big Foot follow a square path. The
maneuver is simply 36 forward steps, followed by a $90^{\circ}$ left turn. The
turn is accomplished by adding $90^{\circ}$ to $\psi_{d}$. The sequence is
repeated until the square is complete. This initial maneuver shows the ability
of Big Foot to make sharp right turns.

The second maneuver is a circular path. Here, Big Foot takes 80 steps. This
maneuver is realized by taking alternating steps with
$\psi_{d}=\psi_{d}+4^{\circ}$ in one step and $\psi_{d}=\psi_{d}+5^{\circ}$ in
the next one.

The third maneuver is a figure eight. It is realized similarly to the circle.
Except, after one loop, one accomplishes the second loop by subtracting from
$\psi_{d}$ instead of adding to it.

The final maneuver is a maze. Lego® bricks are used in assembling the maze.
This maneuver is accomplished using right turns and straight walks similar to
the square maneuver.

These maneuvers show the practical potential of hip actuated bipedal
locomotors. The walking direction can be easily modified by inducing hip
rotations. The work in this paper has shown that gait regulation can be
achieved through hip action.

##  9 Conclusion

The work in this paper presents a new magnetically actuated bipedal walker:
Big Foot. Two important questions regarding bipedal locomotion are answered.
First, Big Foot has shown that bipedal locomotion and maneuvering can be
achieved through pure hip actuation. Second, the analysis shows that it is
preferable to used impulsive actuation rather than continuous actuation
because it results in longer step lengths. In addition, the performance of
this walker is also analyzed through simulation. After defining two distinct
actuation schemes; heel strike and continuous pulse wave actuation schemes. It
is found that heel strike actuation achieves better stability, more
predictable gait generation, and greater locomotion velocities. However,
constant pulse wave actuation achieves locomotion on steeper slopes. What is
also observed is that, even though increasing the pulse duration in heel
strike increases the maximum slope, it also requires more magnetic power.
Interestingly more magnetic power in heel strike results in shorter stride
lengths as the pulse duration is increased. The work opens the door for
experimentation with new designs of bipedal walkers using hip actuation
(magnetic and traditional actuation). The work also shows that magnetic
actuation promises to speed up locomotion research as prototypes are no longer
limited by financial resources and production time. Future research includes
studying more complex legged machines and gait patterns. We will explore tasks
that include stair climbing, obstacle avoidance, and closed-loop control.

We would like to thank Kenny Sangston and Necdet Yildirimer for providing
machining services and space in manufacturing Big Foot. We would also like to
thank The SMU Deason Innovation Gymnasium, led by JT Ringer and Seth Orsborn,
for allowing us to use their laser cutter. Finally, we are acknowledging Ying-
Chu Chen for her work in the development of prototypes that led to the idea of
hip actuation.

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##  10 Appendix A: Index to multimedia Extensions

Table of Multimedia Extensions.

Extension | Media Type | Description
---|---|---
1 | Video | Walking Uphill Increasing Slopes
2 | Video | Walking a Square Path
3 | Video | Walking a Circular Path
4 | Video | Walking a Figure-8
5 | Video | Navigating a Lego® Maze