Dynamics-Based Motion Planning for a Pendulum-Actuated Spherical Rolling Robot

This paper deals with the dynamics and motion planning for a spherical rolling robot with a pendulum actuated by two motors. First, kinematic and dynamic models for the rolling robot are introduced. In general, not all feasible kinematic trajectories of the rolling carrier are dynamically realizable. A notable exception is when the contact trajectories on the sphere and on the plane are geodesic lines. Based on this consideration, a motion planning strategy for complete reconfiguration of the rolling robot is proposed. The strategy consists of two trivial movements and a nontrivial maneuver that is based on tracing multiple spherical triangles. To compute the sizes and the number of triangles, a reachability diagram is constructed. To define the control torques realizing the rest-to-rest motion along the geodesic lines, a geometric phase-based approach has been employed and tested under simulation. Compared with the minimum effort optimal control, the proposed technique is less computationally expensive while providing similar system performance, and thus it is more suitable for real-time applications.     

How to control Chaplygin’s sphere using rotors

In the paper we study the control of a balanced dynamically non-symmetric sphere with rotors. The no-slip condition at the point of contact is assumed. The algebraic controllability is shown and the control inputs that steer the ball along a given trajectory on the plane are found. For some simple trajectories explicit tracking algorithms are proposed.     

Control of the motion of a helical body in a fluid using rotors

This paper is concerned with the motion of a helical body in an ideal fluid, which is controlled by rotating three internal rotors. It is proved that the motion of the body is always controllable by means of three rotors with noncoplanar axes of rotation. A condition whose satisfaction prevents controllability by means of two rotors is found. Control actions that allow the implementation of unbounded motion in an arbitrary direction are constructed. Conditions under which the motion of the body along an arbitrary smooth curve can be implemented by rotating the rotors are presented. For the optimal control problem, equations of sub-Riemannian geodesics on SE(3) are obtained.     

The interconnection of sliding and rolling in the problem of the motion of a homogeneous sphere on a rough horizontal plane

The motion of a homogeneous sphere on a rough horizontal plane when the angular velocities of the twisting and spinning of the sphere are equal to zero at the initial instant is considered. It is proved that, for any initial conditions, the angular velocity of the rolling of the sphere and the sliding velocity vanish after the same finite time. It is shown that the sliding and rolling are interconnected and, in particular, that the rolling of a sphere without sliding is impossible.     

Study on feet forces' distributions,energy consumption and dynamic stability measure of hexapod robot during crab walking

This paper deals with the development of a dynamical model related to crab walking of a hexapod robot to determine the feet forces' distributions, energy consumption and dynamic stability measure considering the inertial effects of the legs on the system, which has not been attempted before. Both forward and inverse kinematic analyses of the robot are carried out with an assigned fixed global frame and subsequent local frames in the trunk body and joints of each leg. Coupled multi-body dynamic model of the robot is developed based on free-body diagram approach. Optimal feet forces and corresponding joint torques on all the legs are determined based on the minimization of the sum of the squares of joint torques, using quadratic programming (QP) method. An energy consumption model is developed to determine the minimum energy required for optimal values of feet forces. To ensure dynamically stable gaits, dynamic gait stability margin (DGSM) is determined from the angular momentum of the system about the supporting edges. Computer simulations have been carried out to test the effectiveness of the developed dynamic model with crab wave gaits on a banking surface. It is observed that when the swing leg touches the ground, impact forces (sudden shoot outs) are generated and their effects are also observed on the joints of the legs. The effects of walking parameters, namely trunk body velocity, body stroke, leg offset, body height, crab angle etc. on power consumption and stability during crab motion for duty factors (DFs) like 1/2, 2/3, 3/4 have also been studied.     

Rolling of a non-homogeneous ball over a sphere without slipping and twisting

Consider the problem of rolling a dynamically asymmetric balanced ball (the Chaplygin ball) over a sphere. Suppose that the contact point has zero velocity and the projection of the angular velocity to the normal vector of the sphere equals zero. This model of rolling differs from the classical one. It can be realized, in some approximation, if the ball is rubber coated and the sphere is absolutely rough. Recently, J. Koiller and K. Ehlers pointed out the measure and the Hamiltonian structure for this problem. Using this structure we construct an isomorphism between this problem and the problem of the motion of a point on a sphere in some potential field. The integrable cases are found.      

Mathematical modeling and trajectory planning of mobile manipulators with flexible links and joints

This paper is concerned with mathematical modeling and optimal motion designing of flexible mobile manipulators. The system is composed of a multiple flexible links and flexible revolute joints manipulator mounted on a mobile platform. First, analyzing on kinematics and dynamics of the model is carried out then; open-loop optimal control approach is presented for optimal motion designing of the system. The problem is known to be complex since combined motion of the base and manipulator, non-holonomic constraint of the base and highly non-linear and complicated dynamic equations as a result of the flexible nature of both links and joints are taken into account. In the proposed method, the generalized coordinates and additional kinematic constraints are selected in such a way that the base motion coordination along the predefined path is guaranteed while the optimal motion trajectory of the end-effector is generated. This method by using Pontryagin’s minimum principle and deriving the optimality conditions converts the optimal control problem into a two point boundary value problem. A comparative assessment of the dynamic model is validated through computer simulations, and then additional simulations are done for trajectory planning of a two-link flexible mobile manipulator to demonstrate effectiveness and capability of the proposed approach.     

Energy-Optimal Multi-Goal Motion Planning for Planar Robot Manipulators

In this work, the energy-optimal motion planning problem for planar robot manipulators with two revolute joints is studied, in which the end-effector of the robot manipulator is constrained to pass through a set of waypoints, whose sequence is not predefined. This multi-goal motion planning problem has been solved as a mixed-integer optimal control problem in which, given the dynamic model of the robot manipulator, the initial and final configurations of the robot, and a set of waypoints inside the workspace of the manipulator, one has to find the control inputs, the sequence of waypoints with the corresponding passage times, and the resulting trajectory of the robot that minimizes the energy consumption during the motion. The presence of the waypoint constraints makes this optimal control problem particularly difficult to solve. The mixed-integer optimal control problem has been converted into a mixed-integer nonlinear programming problem first making the unknown passage times through the waypoints part of the state, then introducing binary variables to enforce the constraint of passing once through each waypoint, and finally applying a fifth-degree Gauss–Lobatto direct collocation method to tackle the dynamic constraints. High-degree interpolation polynomials allow the number of variables of the problem to be reduced for a given numerical precision. The resulting mixed-integer nonlinear programming problem has been solved using a nonlinear programming-based branch-and-bound algorithm specifically tailored to the problem. The results of the numerical experiments have shown the effectiveness of the approach.     

Experimental investigations of the controlled motion of a screwless underwater robot

In this paper we describe the results of experimental investigations of the motion of a screwless underwater robot controlled by rotating internal rotors. We present the results of comparison of the trajectories obtained with the results of numerical simulation using the model of an ideal fluid.     

Omnidirectional walking of legged robots with a failed leg

This paper studies omnidirectional walking of a hexapod robot with a locked joint failure by proposing crab gaits and turning gaits. Due to the reduced workspace of a failed leg, fault-tolerant gaits have limitations in their mobility. As for crab gaits, an accessible range of the crab angle is derived for a given configuration of the failed leg. As for turning gaits, the conditions on turning trajectories guaranteeing fault tolerance are derived for spinning gaits and circling gaits. Based on the principles of fault-tolerant gait planning, periodic crab gaits and turning gaits are proposed in which a hexapod robot realizes tripod walking after a locked joint failure, having a reasonable stride length and stability margin. The proposed fault-tolerant gaits are then applied to an obstacle avoidance problem of a hexapod robot with a locked joint failure. The kinematic constraints of fault-tolerant gaits should be considered in planning the robot trajectory.     

Design an interactive virtual physics environment with uncertainties control to support motion tracking

The development of robot or character motion tracking algorithms is inherently a challenging task. This is more than ever true when the latest trends in motion tracking are considered. Some researchers can deal with kinematic and dynamic constraints induced by the mechanical structure. Another class of researchers fulfills various types of optimality conditions, yet others include means of dealing with uncertainties about the robot or character and its environment. In order to deal with the complexity of developing motion tracking algorithms, it is proposed in this paper to design an interactive virtual physics environment with uncertainties for motion tracking based on sliding mode control. The advantages of doing so are outlined and a virtual environment presented which is well suited to support motion tracking development. The environment makes full use of multi-body system dynamics and a robust sliding mode controller independent of model as simulation kernel. So the environment is capable of simulating setups which fulfill the requirements posed by state-of-the-art motion tracking algorithm development. The demonstration results verified the validity of the environment.     

Trajectory planning based on non-convex global optimization for serial manipulators

To perform specific tasks in dynamic environments, robots are required to rapidly update trajectories according to changing factors. A continuous trajectory planning methodology for serial manipulators based on non-convex global optimization is presented in this paper. First, a kinematic trajectory planning model based on non-convex optimization is constructed to balance motion rapidity and safety. Then, a model transformation method for the non-convex optimization model is presented. In this way, the accurate global solution can be obtained with an iterative solver starting from arbitrary initializations, which can greatly improve the computational accuracy and efficiency. Furthermore, an efficient initialization method for the iterative solver based on multivariable-multiple regression is presented, which further speeds up the solution process. The results show that trajectory planning efficiency is significantly enhanced by model transformation and initialization improvement for the iterative solver. Consequently, real-time continuous trajectory planning for serial manipulators with many degrees of freedom can be achieved, which lays a basis for performing dynamic tasks in complex environments.     

Optimal control of rigid body motion with the help of rotors using stereographic coordinates

This paper considers the problem of optimal controlling the rotational motion of a rigid body using three independent control torques developed by three rotors attached with the principal axes of inertia of the body and rotate with the help of electric motors rigidly mounted on the body. The optimal control law is given as non-linear function of new parameterizations of the rotation group derived by using the stereographic projection of the Euler parameters. Given a cost function we seek for a stabilizing feedback control law that minimizes this cost and asymptotically stabilizes the rotational motion of the body. The stabilizing properties of the proposed controllers are proved by using the optimal Liapunov function. Numerical examples and simulation study are presented.     

Effect of the deformability of structural links on the motion of a two-legged walking robot

The problem of the dynamics of parametric optimization of the motion of a walking robot is solved within the framework of a seven-link model with the elasticity of the structural components taken into account. The effect of the elasticity of the links on the kinematic, dynamic, and energy characteristics of the locomotion of the walking mechanism is studied.Translated from Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 25, pp. 76–79, 1987.     

Navigation of underwater robot based on dynamically adaptive harmony search algorithm

The current research work has employed an evolutionary based novel navigational strategy to trace the collision free near optimal path for underwater robot in a three-dimensional scenario. The population based harmony search algorithm has been dynamically adapted and used to search next global best pose for underwater robot while obstacle is identified near about robot’s current pose. Each pose is evaluated based on their respective value for objective function which incorporates features of path length minimization as well as obstacle avoidance. Dynamic adaptation of control parameters and new perturbation schemes for solution vectors of harmony search has been proposed to strengthen both exploitation and randomization ability of present search process in a balanced manner. Such adaptive tuning process has found to be more effective for avoiding early convergence during underwater motion in comparison with performances of other popular variants of Harmony Search. The proposed path planning method has also shown better navigational performance in comparison with improved version of ant colony optimization and heuristic potential field method for avoiding static obstacles of different shape and sizes during underwater motion. Simulation studies and corresponding experimental verification for three-dimensional navigation are performed to check the accuracy, robustness and efficiency of proposed dynamically adaptive harmony search algorithm.     

Integrable discretization and deformation of the nonholonomic Chaplygin ball

The rolling of a dynamically balanced ball on a horizontal rough table without slipping was described by Chaplygin using Abel quadratures. We discuss integrable discretizations and deformations of this nonholonomic system using the same Abel quadratures. As a by-product one gets a new geodesic flow on the unit two-dimensional sphere whose additional integrals of motion are polynomials in the momenta of fourth order.     

Solving Optimal Control Problems by Exploiting Inherent Dynamical Systems Structures

Computing globally efficient solutions is a major challenge in optimal control of nonlinear dynamical systems. This work proposes a method combining local optimization and motion planning techniques based on exploiting inherent dynamical systems structures, such as symmetries and invariant manifolds. Prior to the optimal control, the dynamical system is analyzed for structural properties that can be used to compute pieces of trajectories that are stored in a motion planning library. In the context of mechanical systems, these motion planning candidates, termed primitives, are given by relative equilibria induced by symmetries and motions on stable or unstable manifolds of e.g. fixed points in the natural dynamics. The existence of controlled relative equilibria is studied through Lagrangian mechanics and symmetry reduction techniques. The proposed framework can be used to solve boundary value problems by performing a search in the space of sequences of motion primitives connected using optimized maneuvers. The optimal sequence can be used as an admissible initial guess for a post-optimization. The approach is illustrated by two numerical examples, the single and the double spherical pendula, which demonstrates its benefit compared to standard local optimization techniques.     

Path planning for robots under stochastic uncertainty *

In order to reduce large online measurement and correction expenses, the a priori informations on the random variations of the model parameters of a robot and its working environment are taken into account already at the planning stage. Thus, instead of solving a deterministic path planning problem with a fixed nominal parameter vector, here, the optimal velocity profile along a given trajectory in work space is determined by using a stochastic optimization approach. Especially, the standard polygon of constrained motion-depending on the nominal parameter vector-is replaced by a more general set of admissible motion determined by chance constraints or more general risk constraints. Robust values (with respect to stochastic parameter variations) of the maximum, minimum velocity, acceleration, deceleration, resp., can be obtained then by solving a univariate stochastic optimization problem Considering the fields of extremal trajectories, the minimum-time path planning problem under stochastic uncertainty can be solved now by standard optimal deterministic path planning methods     

Time Optimal Path Planning for Industrial Robots: A Dynamic Programming Approach Considering Torque Derivative and Jerk Constraints

This paper presents a dynamic programming approach for calculating time optimal trajectories for industrial robots, subject to various physical constraints. In addition to path velocity, motor torque, joint velocity and acceleration constraints, the present contribution also shows how to deal with torque derivative and joint jerk limitations. First a Cartesian path for the endeffector is defined by splines using Bernstein polynomials as basis functions and is parameterized via a scalar path parameter. In order to compute the belonging quantities in configuration space, inverse kinematics is solved numerically. Using this and in combination with the dynamical model, joint torques as well as their derivatives can be constrained. For that purpose the equations of motion are calculated with the help of the Projection Equation. As a consequence of the used optimization problem formulation, the dynamical model as well as the restrictions have to be transformed to path parameter space. Due to the additional consideration of jerk and torque derivative constraints, the phase plane is expanded to a phase space. The parameterized restrictions lead to feasible regions in this space, in which the optimal solution is sought. Result of the optimization is the time behavior of the path parameter and subsequently the feed forward torques for the optimal motion on the spatial path defined by previously mentioned splines. Simulation results as well as experimental results for a three axes industrial robot are presented. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Motion equations of cooperative multi flexible mobile manipulator via recursive Gibbs–Appell formulation

In this paper, the developed model of an N-flexible-link mobile manipulator with revolute-prismatic joints is presented for the cooperative flexible multi mobile manipulator. In this model, the deformation of flexible links is calculated by using the assumed modes method. In additions, non-holonomic constraints of the robots’ mobile platforms that bound its locomotion are considered. This limitation is alleviated through the concurrent motion of revolute and prismatic joints, although it results in computational complexity and changes the final motion equations to time-varying form. Not only is the proposed dynamic model implemented for the multi-mobile manipulators with arms having independent motion, but also for multi-mobile manipulators in cooperation after defining gripper's kinematic constraints. These constraints are imported to the dynamic equations by defining Lagrange multipliers. The recursive Gibbs–Appell formulation is preferred over other similar approaches owing to the capability of solving the equations without the need to use Lagrange multipliers for eliminating non-holonomic constraints in addition to the novel optimized process of obtaining system equations. Hence, cumbersome simultaneous computations for eliminating the constraints of platform and arms are circumvented. Therefore, this formulation is improved for the first time by importing Lagrange multipliers for solving kinematic constrained systems. In the simulation section, the results of forward dynamics solution for two flexible single-arm manipulators with revolute-prismatic joints while carrying a rigid object are presented. Inverse dynamics equations of the system are also presented to obtain the maximum dynamic load-carrying capacity of the two-rigid-link mobile manipulators on a predefined path. Two constraints, namely the capacity of joint motors torque and robot motion stability are considered as the limitation criteria. The concluded motion equations are used to accurately control the movement of sensitive bodies, which is not achievable through the use of one platform.