Model-based Locomotion Control of Underactuated Snake Robots
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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 . 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 . 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.