## Model-based Locomotion Control of Underactuated Snake Robots

##### Abstract

Snake robots are a class of biologically inspired robots which are built
to emulate the features of biological snakes. These robots are underactuated,
i.e. they have fewer control inputs than degrees of freedom, and are
hyper redundant, i.e. they have many degrees of freedom. Furthermore,
snake robots utilize complex motion patterns and possess a complicated
but highly flexible physical structure. These properties make locomotion
control of snake robots a complicated and challenging control problem.
This thesis considers model-based locomotion control of planar snake
robots. In particular, based on kinematic and dynamic models of the snake
robot locomotion, using different control approaches we derive feedback
control laws in order to solve various control problems. Moreover, through
rigorous mathematical stability analysis, we prove the stability of the controlled
system. It is noteworthy to mention that due to the complicated dynamical
behavior of snake robots which gives rise to a complex dynamic
model, and also the underactuation which is characterized by the lack of
direct and independent control inputs for at least three degrees of freedom
of the snake robot, the vast majority of the previous works on snake robots
and similar multi-link robotic structures use numerical simulations and
experimental results which are obtained using different robotic snakes, as
the main tools to show the performance of the proposed controllers. In
contrast, however, in this work based on nonlinear control theory, we take
a model-based control design approach and we present formal stability
proofs for the closed-loop systems along with numerical simulations and
experimental results. The simulations and experiments are performed for
a snake robot which is composed of N similar links which are serially connected
through N - 1 joints. The first N - 1 links are independently actuated
using electric motors, however, the N-th link which we refer to as
the head link of the snake robot is passive. This makes the orientation and
position of the center of mass of the robot underactuated.
The contributions of the thesis are presented in six chapters, and can
be categorized in two types; contributions to modelling and contributions
to control design for snake robots. The contributions and contents of each
chapter are as follows.
In Chapter 1, we discuss the fundamental properties of the snake robot
locomotion, and we investigate the most common types of gait patterns
used by biological snakes. Furthermore, in this chapter we review the
relevant previous works on snake robots and we present the abstracts of the academic papers which form the basis of the thesis.
In Chapter 2, we present three different modelling techniques for the
snake robot locomotion on horizontal and flat surfaces. The first dynamic
model is derived based on the Lagrangian approach to modelling mechanical
systems, and the equations of motion are written in the standard
second-order form. The second dynamic model is derived using the
techniques of differential geometry, and this model contains the effects of
parametric modelling uncertainties on the locomotion of the robot. The
first and the second models which are referred to as the complex model of
the snake robot, are among the contributions of the thesis and to our best
knowledge have not been presented in any previous works. The third
model that we present in Chapter 2 is a simplified model of the snake
robot locomotion which is previously presented in [11]. In this simplified
model, the rotational motion of the joints is mapped to translational
link displacements. Through this mapping, which is shown to be valid for
small joint angles, many of the strong nonlinear terms which are present
in the dynamics of the system are approximated by simpler linear terms,
and these approximations make the resulting simplified dynamic model
more amenable to model-based control design.
In Chapter 3, we consider body shape and orientation control for locomotion
of snake robots. In particular, in this chapter we aim to control
the body shape of the robot to a desired gait pattern, and the orientation of
the robot to a reference angle defined by a path following guidance law. To
this end, using the joint torques we stabilize a desired gait pattern among
the directly actuated body shape variables which define the internal configuration
of the robot. Furthermore, we use a gait parameter in the form
of a dynamic compensator which controls the orientation of the robot to a
reference angle defined by the path following guidance law. Through numerical
simulations and experiments which are performed using a robotic
snake, we show that this control approach makes the robot converge to
and follow a desired geometric path. Moreover, using an input-output stability
analysis we show that the solutions of the controlled system remain
uniformly bounded. Furthermore, in this chapter using sliding mode techniques,
we design a body shape and orientation feedback controller which
successfully makes the robot follow a desired path even in the presence of
strong nonlinear terms in the dynamics of the robot arising due to parametric
modelling uncertainties.
In Chapter 4, we utilize the simplified model of the snake robot locomotion
to carry out the model-based feedback control design for the robot.
In particular, we use the method of virtual holonomic constraints (VHC) to
address direction following and maneuvering control of the snake robot.
In the direction following problem, the control objective is to regulate the
linear velocity vector of the snake robot to a constant reference while guaranteeing
boundedness of the system states. Furthermore, in the maneuvering
problem, the control objective is to make the robot converge to a
geometric path, and to move along the path according to a desired velocity
profile. Using the VHC method, we stabilize the solutions of the
dynamics of the robot to a constraint manifold. The constraint manifold
is defined based on VHC which encode a lateral undulatory gait pattern.
Moreover, this gait pattern is parameterized by the states of two dynamic
compensators which are used to control the forward velocity and orientation
of the robot.
In Chapter 5, we utilize the complex model of the snake robot locomotion
in order to address the direction following and maneuvering control
problems. In particular, first we stabilize VHC for the body shape variables
of the system which encode a lateral undulatory gait pattern. The
VHC are composed of a sinusoidal part and an offset term. The sinusoidal
part is employed to induce the lateral undulatory motion and the offset
term is used to reorient the robot in the plane. Furthermore, the VHC are
dynamic in that they depend on the states of two dynamic compensators
which are used in order to control the forward velocity and orientation
of the robot. In particular, using a high-gain feedback on the offset term,
we turn the controlled orientation dynamics of the robot into a singularly
perturbed form, for which we show that the orientation error can be made
arbitrarily small. In addition, we use the frequency of the oscillations of
the snake body, i.e. the frequency of the desired gait pattern, as a virtual
control input which is used to control the forward velocity of the robot.
Using backstepping techniques, we make the forward velocity error arbitrarily
small and make the normal velocity converge to a small neighborhood
of zero. This solves the direction following problem. In order to address
the maneuvering problem using VHC, we use a hierarchical control
approach based on a reduction theorem for asymptotic stability of dynamical
systems presented in [98]. In particular, first we stabilize a constraint
manifold for the robot and then we control the reduced dynamics of the
robot on the constraint manifold using two dynamic compensators. These
dynamic compensators control the forward velocity and the head angle of
the robot to given references. Furthermore, we define the reference head
angle and the reference velocity of the robot such that the convergence of
the path following error to an arbitrarily small neighborhood of the origin
is guaranteed. Extensive numerical simulations are presented which validate
the performance of the proposed control strategies.
Finally, in Chapter 6, we summarize the contributions of the chapters
and present some concluding remarks. Furthermore, we present topics for
possible future works on locomotion control of snake robots.