Reactive Self-Collision Avoidance for a Differentially Driven Mobile Manipulator

This paper introduces a reactive self-collision avoidance algorithm for differentially driven mobile manipulators. The proposed method mainly focuses on self-collision between a manipulator and the mobile robot. We introduce the concept of a distance buffer border (DBB), which is a 3D curved surface enclosing a buffer region of the mobile robot. The region has the thickness equal to buffer distance. When the distance between the manipulator and mobile robot is less than the buffer distance, which means the manipulator lies inside the buffer region of the mobile robot, the proposed strategy is to move the mobile robot away from the manipulator in order for the manipulator to be placed outside the border of the region, the DBB. The strategy is achieved by exerting force on the mobile robot. Therefore, the manipulator can avoid self-collision with the mobile robot without modifying the predefined motion of the manipulator in a world Cartesian coordinate frame. In particular, the direction of the force is determined by considering the non-holonomic constraint of the differentially driven mobile robot. Additionally, the reachability of the manipulator is considered to arrive at a configuration in which the manipulator can be more maneuverable. In this respect, the proposed algorithm has a distinct advantage over existing avoidance methods that do not consider the non-holonomic constraint of the mobile robot and push links away from each other without considering the workspace. To realize the desired force and resulting torque, an avoidance task is constructed by converting them into the accelerations of the mobile robot. The avoidance task is smoothly inserted with a top priority into the controller based on hierarchical quadratic programming. The proposed algorithm was implemented on a differentially driven mobile robot with a 7-DOFs robotic arm and its performance was demonstrated in various experimental scenarios.

Formulation and solution of a generalized Chebyshev problem. II

This work is a continuation of the article “Formulation and solution of a generalized Chebyshev problem. I”, in which a generalized Chebyshev problem was formulated, two theories of motion for non-holonomic systems with high-order constraints were presented for its solution. These theories were used to study the motion of the Earth’s satellite when fixing the magnitude of its acceleration (this was equivalent to imposing a linear non-holonomic constraint of the third order). In the offered article, the second theory, based on the application of the generalized Gauss principle, is used to solve one of the most important problems of control theory: finding the optimal control force that translates a mechanical system with a finite number of degrees of freedom from one phase state to another in a specified time. The application of the theory is demonstrated by solving a model problem of controlling the horizontal motion of a cart bearing the axes of s mathematical pendulums. Initially, the problem is solved by applying the Pontryagin maximum principle, which minimizes the functional of the square of the desired horizontal control force, which transfers the mechanical system from a state of rest to a new state of rest in the specified time with the horizontal displacement of the cart by S (that is, the problem of vibration damping is considered). Let’s call this approach the first method of solving the control problem. It is shown that a linear non-holonomic constraint of the order (2s + 4) is continuously performed. This suggests applying the second theory of motion for non-holonomic systems with high-order constraints to the same problem (see the previous article), developed at the Department of Theoretical and Applied Mechanics of the Faculty of Mathematics and Mechanics of Saint Petersburg State University. Let’s call this approach the second method of solving the problem. Calculations performed for the case of s = 2 showed that the results obtained by the both methods are practically the same for a short-term motion of the system, while they differ sharply for a long-term motion. This is because the control found using the first method contains harmonics with the system’s natural frequencies, which tends to bring the system into resonance. With a short-term motion, this is not noticeable, and with a long-term motion, there are large fluctuations in the system. In contrast, when using the second method, the control is in the form of a polynomial in time, which provides a relatively smooth motion of the system. In addition, in order to eliminate the control force jumps at the beginning and end of the motion, we propose to solve a generalized boundary value problem and discuss some special cases that sometimes occur when using the second method for solving the boundary value problem.