Motion planning for a planar mechanical system with dissipative forces
Introduction
To overcome the limits of wheeled robots on uneven terrain, more versatile locomotion mechanisms inspired by nature have gained the attention of many robotics research groups. New designs for legged, flying, or crawling robots have been found to be more effective for specific applications. A particularly effective structure for mobility in confined spaces such as pipe inspection, collapsed buildings, and mine shafts is the snake robot. Hirose and colleagues pioneered the field of snake robotics by introducing an active cord mechanism for snake-like motion [1]. Mechanical designs of snake robots and the serpenoid curve were achieved through extensive experimentation on snakes for generating gaits to imitate a slithering motion [[1], [2]]. Snake models built by many research groups are either wheeled to realize the kinematic non-holonomic constraints during the slithering motion [3] or have a un-wheeled modular design to comprehend real snakes [[4], [5]].
Since a robotic locomotion system does not have direct control over position and orientation in an inertial frame, it is an under-actuated system. For this type of system, geometric mechanics provides a structured way to define reduced ordered Lagrangian dynamic model by representing system dynamics in the local body frame [[6], [7], [8]]. Geometric motion planning exploits the structure of the reduced order dynamic model of the locomotion system to design gaits to change position in ambient space [[6], [9], [10], [11]]. The mobility of the robot can be achieved by cyclic changes in the shape of the robot [[7], [12], [13]] as observed in locomotion mechanisms existing in nature.
In the absence of external forces, mechanical locomotion systems without kinematic nonholonomic constraints are purely mechanical systems [14]. Such systems possess conserved quantities as Lagrangian symmetries such as linear or angular momentum. Such quantities are considered as dynamic non-holonomic constraints [[6], [15]]. The locomotion mechanics of such systems can be modeled by mechanical connection on the principle fiber bundle of the configuration space. However, for the above-mentioned class of mechanical systems in the absence of interaction with the environment, dynamic nonholonomic constraints limit mobility in ambient space. In contrast, friction forces due to interaction with an environment enable the system to move along the constrained directions. In [[16], [17]], the effect of friction forces on the dynamics of a 2-link robot moving on the ground using dry friction was considered. The method used to generate motion was composed of the slow and fast motion of the rear smaller link. However, the motion of the systems is constrained with the difference in sizes of link lengths. The geometry of the viscous dissipative forces on the principle bundle structure of the configuration space can be represented by Stoke’s Connection. In [18], Stoke’s Connection was used to study the controllability of the deformable bodies in a viscous environment. In this paper, we introduce the Stoke’s Height Function to represent the curvature of the Stoke’s Connection. The Stoke’s Height Function is utilized to quantify the net change in momentum along a fiber direction, in response to viscous dissipative forces, and the gait selection.
A generalized geometric motion planning framework was proposed by Shammas et al. in [[10], [11], [14]] for different classes of systems with spectrum of kinematic and dynamic nonholonomic constraints. However, this method does not consider the effect of the external forces, like dissipative forces, which are the main agent enabling the systems without kinematic nonholonomic constraints to move in ambient space. This paper highlights how the effect of dissipative forces can be considered in the geometric motion planning and gait selection. In order to quantify the dynamic phase shift, we need to evaluate the momentum evolution equation in the reduced order dynamics of the system. In [11], scaled momentum was introduced to simplify the momentum evolution equation and evaluate the dynamic phase shift. However, this simplification is conditioned by the existence of the integrating factor for the momentum evolution equation. This simplification has limitation that integrating factor and scaled momentum were presented with the assumption that generalized momentum is one dimensional. However, there is no general way to find integrating factor and hence scaled momenta terms for simultaneous coupled nonlinear differential equations with more than one generalized momenta as in our case. Alternatively, our work exploits the geometric structure of the dissipative forces by defining the Stoke’s Height function and Stoke’s Gamma function to quantify dynamic phase shift. This helps evaluate explicitly and easily the effects of centrifugal and Coriolis forces, coupling of the generalized momenta and joint (shape) velocities in dynamic phase shift. This work is more consistent with the geometric motion planning framework to intuitively find and evaluate a closed curve gait in joint space) for a system with dissipative forces.
In [19], McIsaac et al. investigated the motion planning problem for a body moving in high Reynolds number. However, they ignored the longitudinal friction forces and also the coupling between the generalized momenta. In contrast, our proposed method considers both aspects in the system dynamic equations. Furthermore, McIsaac et al. in [19] evaluated various gaits. However, gait selection is not intuitive and the trajectory followed by system in the shape space to execute the motion in ambient space was not considered. To overcome this problem, our work addresses both aspects of the motion planning.
Motion planning problem for a two link system moving in viscous medium was also studied by introducing friction pads, and the gait design method was proposed in [20]. The geometric structure of the dynamic forces are represented explicitly as push, drift, and cross terms in the dynamic model of the system. However, the gait design method is represented only for two link system with single degree of freedom. Furthermore, this gait design method does not give direct intuition about gait selection but using the constraints on the push and cross terms to nullify motion in undesired direction. This paper represents similar structure of the viscous dissipative forces as Dissipative Gamma Functions matrix, Dissipative Connection matrix and Centrifugal, and Coriolis terms, respectively. Furthermore, motion planning using the geometric structure properties of the dissipative forces, as proposed in this paper is easier and more intuitive.
In this paper, we address the geometric motion planning problem for a snake robot moving on the ground. For the sake of simplicity, each link is assumed to have point contact with the environment under conditions of frictional anisotropy. This assumption can be realized by an elongated oval shape of the link with point contact with the ground during motion [21]. The link profile is responsible for maintaining the anisotropic nature of the friction model due to different coefficients of friction along longitudinal and lateral axes for each link. The properties of the geometric structure of dissipative forces along with that of system dynamics are also exploited to choose appropriate gaits for an unconstrained 3-link snake model moving on a flat surface.
This paper makes three major contributions to the geometric motion planning:
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The first contribution of this article is to analyze the effect of dissipative forces in reduced order dynamics using the Stoke’s Height function and Stoke’s Gamma functions. These functions can be formulated by integrating the momentum evolution equation using Stoke’s Theorem. These functions are helpful in quantifying the change in momentum for a closed curve gait.
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Second, we examine how the coupling between the generalized momenta can be analyzed intuitively using Stoke’s Gamma functions.
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Third, using Stoke’s connection and Stoke’s Gamma functions, the equations of motion with the dissipative forces as external forces can be analyzed easily for motion planning problem. Gait selection becomes simple and intuitive by analyzing properties of the Stokes’ Height Functions and Stokes’ Gamma Functions along with Height Functions and Gamma Functions from reconstruction equation rather than solving the differential equations using integrating factor (which may exist but is difficult to find).
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Fourth, the proposed methodology is demonstrated experimentally, which supports the simulation results.
The structure of this article is as follows. Section 2 outlines a summary of the reduced order dynamics based on Lagrangian mechanics. Section 3 details the dynamic model formulation for the system used in this paper and how the conserved quantities due to Lagrangian symmetries limit the evolution of motion in some directions. The friction model with its geometric structure is presented in Section 4. Section 5 provides the evaluation of the robot motion using reduced order dynamics model of the system for a given closed curve gait in base space. In Section 6, gait selection rules are described, which are based on exploiting the geometric structure of a system’s reduced dynamics along with that of the friction model. The simulation tests and experimental demonstration are given in Sections 7 Simulation tests, 8 Experimental Demonstration, respectively. Finally, a discussion on results and proposed future work in Section 9.
Section snippets
Background
This section presents a brief overview of Lagrangian mechanics of mechanical locomotion systems with Lagrangian symmetry. Generally, -dimensional configuration space of a locomotion mechanical system is manifold with a trivial principle fiber bundle structure. It can be written as the product of two subspaces where is -dimensional fiber space that represents the position and orientation of the system with respect to the inertial frame and possesses aLie group structure. Also,
Reduced order dynamic model of the system
In this section, we present the dynamic model of the three link snake robot. The purpose of this model design is to realize a simplified snake on a flat surface. Each link has an elongated oval shape to implement point contact with less drag along the longitudinal axis than the lateral axis. Each link has a length of , the moment of inertia and the mass acting at the center of the link. The orientation of each link with respect to the inertial frame -axis is defined as . The local body
Friction model
Terrestrial Snakes rely upon dry solid–solid friction for propulsion [30]. To reflect this, Coulomb’s Friction model is used to model dry friction force which is proportional to the normal force at contact with the opposite sign of the velocity of the body . where is the coefficient of friction and the represents the sign () of the function or variable in the input argument. However, due to the discontinuous nature of Coulomb’s friction model in the presence of the
Cyclic gait and robot motion
Animals interact with the environment to generate motion by continuously changing their shape cyclically by appropriately actuating their joints [[7], [20], [25]]. To exploit the cyclic changes in shape to steer a mechanical system, sinusoidal inputs were proposed in [[31], [32]] as these are easy to design. Locomotion system moving in ambient space (SE(2) for the planar case) is considered under-actuated, as there is no direct control over the fiber variables. Despite this weakness, it can
Gait selection
Gait selection method presented in this paper is based on the guidelines presented in [[10], [14]] to define a rule base using the height functions, and gamma function. However, we can also use the same guidelines for dissipative height and dissipative gamma functions in a friction model to plan motion for a purely mechanical system with dissipative contact with environment. The key functions that play a significant role in the motion planning method are
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For the geometric phase
Simulation tests
This section demonstrates three gait test results and gait evaluation for motion along one or more fiber directions. In this section, the simulation results were obtained using MatLab ® 2016. The parameters used were . The resultant motion along the linear -, and -axes are represented in terms of , and rotation is represented in radians.
Experimental Demonstration
For experimental demonstration, we developed a prototype model of 2-dof planar robotic locomotion system shown in Fig. 9. Links of the prototype are fabricated using 3D printer with PLA material. A dummy weight is placed over middle link such that mass of each link is identical. Each link interacts with ground through rubber pads (coefficients of friction ). Two Dynamixel RX-28 servos are used to independently drive the joints through timing belts and pulleys. The actuators are controlled
Discussion and future work
In this paper, the role of dissipative forces is evaluated in the dynamics and motion planning of a three-link snake robot moving on the ground. The dissipation forces are generated due to system interaction with the environment. To model the ground contact dissipation forces, a viscous friction model is used rather than Coulomb’s friction model with a discontinuity. By using the geometric mechanics tools, the viscous friction model can be written in a structured way as a linear function of the
Acknowledgments
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education ( 2015R1D1A1A01060741). Also, the authors would like to thank the Higher Education Commission (HEC) of Pakistan for scholarship Program “HRDI for faculty development for UESTPS”.
Ahmad Ali received B.Sc. Engineering Degree in Mechatronics and Control Engineering in 2006 from University of Engineering & Technology, Lahore, Pakistan and MS Degree in Automation and Control Technologies in 2011 from Politecnico di Torino, Italy. Currently, he is a Ph.D. student in CnR lab at Hanyang University, South Korea. His research interests are Systems Analysis, Nonlinear Control Theory, Geometric Control of Mechanical Systems, and Motion Planning for Nonholonomic System.
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Ahmad Ali received B.Sc. Engineering Degree in Mechatronics and Control Engineering in 2006 from University of Engineering & Technology, Lahore, Pakistan and MS Degree in Automation and Control Technologies in 2011 from Politecnico di Torino, Italy. Currently, he is a Ph.D. student in CnR lab at Hanyang University, South Korea. His research interests are Systems Analysis, Nonlinear Control Theory, Geometric Control of Mechanical Systems, and Motion Planning for Nonholonomic System.
Sheraz Yaqub received his B.Sc. degree in Mechatronics Engineering from University of Engineering & Technology (U.E.T.) Taxila, Chakwal campus, Pakistan in 2013. He worked for Sugar Industry as assistant instrumentation and control engineer for two years. He is currently pursuing his master’s leading to Ph.D. in Mechatronics at Hanyang University, Korea. His research interests include robot motion planning, Control Theory, Under-actuated Robots, Embedded Systems.
Muhammad Usman received his BSc degree in Mechatronics and Control Engineering from University of Engineering & Technology (U.E.T.) Lahore, Pakistan in 2008. He worked for an Automation company for three months in 2009. Since September 2009, he has been working as a lecturer in Mechatronics & Control Engineering Department in University of Engineering & Technology Lahore (Faisalabad Campus), Pakistan. He received his MSc Mechatronics degree from National University of Singapore in 2012 and restored his teaching job in U.E.T. He is currently on study leave and pursuing his Ph.D. in Mechatronics at Hanyang University, Korea. His research interests include Autonomous vehicles, SLAM, Data association, Navigation and Control.
Khalil Muhammad Zuhaib received the B.E. degree in Electronics Engineering from Quaid-e-Awam University, Pakistan in 2009. He is currently enrolled in MS-Ph.D. in Mechatronics Engineering at Hanyang University, Korea. His current research interests include multi-robot systems and robot motion planning. He is also interested in working on new ideas in this direction.
Abdul Manan Khan received his B.Sc. and M.Sc. degrees in Mechatronics & Control Engineering from University of Engineering & Technology, Lahore, Pakistan, in 2007 and 2010, respectively, and his Ph.D. degree in Mechanical Design Engineering from Hanyang University, 2016, Korea. His research interests include rehabilitation and assist exoskeleton robots. He has worked in many projects including lower limb and upper limb assist exoskeleton robots. He is also interested in working on motion planning based different task executions and control. He also has expertise in Robotic Operating System (ROS) and C++.
Ji Yeong Lee received his B.Sc. and M.Sc. degrees in Mechanical Engineering from Seoul National University, South Korea, in 1991 and 1993, respectively, and his Ph.D. degree in Mechanical Engineering from Carnegie Mellon University, USA, in 2003. From 1993 to 1998, he was researcher in Automation Research Institute, Samsung Electronics, South Korea. From 2003 to 2007, he was senior researcher in department of Mechanical Engineering, KAIST, South Korea. In 2007, he joined Hanyang University, South Korea. Currently, he is Associate Professor in Department of Robotics Engineering, Hanyang University. His research interests include robotics & control, path planning, Articulated robots and non-holonomic systems.
Changsoo Han received the B.S. degree in Mechanical Engineering from Hanyang University in 1983, and his M.S. and Ph.D. degrees in Mechanical Engineering from University of Texas at Austin in 1985, 1989, respectively. From September 1984 to May 1985, he was a Teaching Assistant with CAD/CAM Lab in the department of engineering of the University of Texas at Austin. From October 1987 to April 1988, he was the consultant for a Lockheed MAC design project for the Lockheed Austin Division. From May 1988 to September 1989, he was a research assistant, Robotics Lab in mechanical engineering manufacturing of thehigh-resolution micromanipulator. He stayed at University of California at Berkeley as a visiting professor from August 1996 to July 1997. In March 1990, he joined Hanyang University, Ansan, Korea as an assistant professor in the department of mechanical engineering. Currently, he is a Professor in the Department of robot engineering, Hanyang University. His research interests include intelligence service robot, high precision robotics and mechatronics, rehabilitation and biomechanics technology using robotics, automation in construction, advanced vehicle control and assistive exoskeleton robots.