A structured algorithm for minimum l_∞-norm solutions and its application to a robot velocity workspace analysis

Abstract

Herein is proposed a concise algorithmic procedure for deriving a minimum l∞-norm solution of the system of consistent linear equations Ax=b, where A is a m×n matrix, b is given as a m×1 vector and x is a n×1 unknown vector. The proposed algorithm is developed based on the geometrical analysis of the characteristics of the minimum infinity-norm solution. The proposed algorithm is well-structured so that it may be implemented easily through simple linear algebraic manipulation. For the case of n>m, the basic idea of the method is extended to finding internally mapped vertices when a n dimensional polytope is transformed into m dimensional polytope through consistent linear mapping A. The proposed method is applied to the task velocity analysis for robot manipulators with joint velocity constraints.

Higher Dimensional Spatial Expression of Upper Limb Manipulation Ability Based on Human Joint Torque Characteristics

“…Even in the field of robotics, the visualization method of a higher-dimensional hyperellipsoid defined in a space with dimensions equal to the degrees of freedom of a hand, has not been clarified yet. A slack variable is generally introduced to search for the vertex of the polytope (Shim & Yoon, 1997;Chiacchio et al, 1997;Lee, 2001). It is extremely difficult to search for the region of a higher-dimensional polytope accurately using these methods because of its huge computational complexity, and it is also difficult to formulate an objective function in the case of linear programming.…”

Bi‐criteria optimal control of redundant robot manipulators using LVI‐based primal‐dual neural network

“…In this section, the LVI-PDNN solver is developed as follows to solve the bi-criteria redundancy-resolution problem (3)-(6), which has been rewritten as QP (8)- (11) in the preceding section (i.e. a more mathematically tractable form).…”

“…The dual decision variable is often defined as the Lagrangian multiplier for each constraint such as (9)- (11). However, to reduce the structural complexity of QP-solvers, an elegant treatment could be used to cancel the dual variables for bound constraint (11).…”

“…The research of recent ten years shows that redundancy-resolution problems might be solved in a more favorable manner via online optimization techniques [4,7,8]. In addition, people resolve the redundancy problem mainly at the level of joint velocities by optimizing various performance criteria [6,[8][9][10][11][12][13][14][15][16][17][18][19]. The optimal control schemes at joint-velocity level could include infinity-norm velocity minimization (INVM) scheme [9-11, 13, 14], minimum velocity norm (MVN) scheme [12,18], minimum kinetic energy (MKE) scheme [7,20,21], bi-criteria velocity minimization scheme [4,10,16], repetitive motion planning scheme [4,8,17], and so on.…”

Vertex Search Algorithm of Convex Polyhedron Representing Upper Limb Manipulation Ability

“…The manipulability of the upper and lower limbs in three-dimensional task space is expressed as an invisible six-dimensional polytope. For such evaluation, a slack variable is generally introduced in order to search for the vertex of the polytope (Shim & Yoon, 1997;Chiacchio et al, 1997;Lee, 2001). However, it is extremely difficult to search for the region of a higher-dimensional polytope accurately using conventional methods because of their huge computational complexity, and it is also difficult to formulate an objective function in the case of linear programming.…”